Systems and methods for determining an initial margin

ABSTRACT

An exemplary system according to the present disclosure comprises a computing device that in operation, causes the system to receive financial product or financial portfolio data, map the financial product to a risk factor, execute a risk factor simulation process involving the risk factor, generate product profit and loss values for the financial product or portfolio profit and loss values for the financial portfolio based on the risk factor simulation process, and determine an initial margin for the financial product. The risk factor simulation process can be a filtered historical simulation process.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of Ser. No. 17/221,052 filed Apr. 2,2021, which is a continuation of Ser. No. 17/104,403 filed Nov. 25, 2020(now U.S. Pat. No. 11,023,978), which is a continuation of Ser. No.16/775,970 filed Jan. 29, 2020 (now U.S. Pat. No. 10,922,755), which isa continuation-in-part of U.S. application Ser. No. 16/046,190 filedJul. 26, 2018 (now U.S. Pat. No. 10,817,947), which is a continuation ofU.S. application Ser. No. 14/303,941 filed Jun. 13, 2014 (now U.S. Pat.No. 10,102,581), which claims the benefit of U.S. Application Ser. No.61/835,711 filed Jun. 17, 2013, the entire contents of each of which areincorporated herein by reference.

TECHNICAL FIELD

This disclosure relates generally to financial products, methods andsystems, and more particularly to systems and methods forcollateralizing risk of financial products.

BACKGROUND

Conventional clearinghouses collect collateral in the form of an“initial margin” (“IM”) to offset counterparty credit risk (i.e., riskassociated with having to liquidate a position if one counterparty of atransaction defaults). In order to determine how much IM to collect,conventional systems utilize a linear analysis approach for modeling therisk. This approach, however, is designed for financial products, suchas equities and futures, that are themselves linear in nature (i.e., theproducts have a linear profit/loss scale of 1:1). As a result, it is notwell suited for more complex financial products, such as options,volatile commodities (e.g., power), spread contracts, non-linear exoticproducts or any other financial products having non-linear profit/lossscales. In the case of options, for example, the underlying product andthe option itself moves in a non-linear fashion, thereby resulting in anexponential profit/loss scale. Thus, subjecting options (or any othercomplex, non-linear financial products) to a linear analysis willinevitably lead to inaccurate IM determinations.

Moreover, conventional systems fail to consider diversification orcorrelations between financial products in a portfolio when determiningan IM for the entire portfolio. Instead, conventional systems simplyanalyze each product in a portfolio individually, with no considerationfor diversification of product correlations.

Accordingly, there is a need for a system and method that efficientlyand accurately calculates IM for both linear and non-linear products,and that considers diversification and product correlations whendetermining IM for a portfolio of products.

SUMMARY

The present disclosure relates to systems and methods of collateralizingcounterparty credit risk for at least one financial product or financialportfolio comprising mapping at least one financial product to at leastone risk factor, executing a risk factor simulation process comprising afiltered historical simulation process, generating product or portfolioprofit and loss values and determining an initial margin for thefinancial product or portfolio.

The present disclosure also relates to systems, methods andnon-transitory computer-readable mediums for efficiently modelingdatasets. A system includes at least one computing device comprisingmemory and at least one processor. The memory stores a margin model anda liquidity risk charge (LRC) model. The processor executescomputer-readable instructions that cause the system to: receive, asinput, data defining risk factor data and additional data associatedwith at least one financial portfolio, where the at least one financialportfolio comprises at least one financial product and one or morecurrencies. The instructions also cause the system to execute the marginmodel, causing the margin model to execute a risk factor simulationprocess involving the received risk factor data. The risk factorsimulation process comprises a filtered historical simulation process.The instructions also cause the system to generate, by the margin model,portfolio profit and loss values for the at least one financialportfolio based on output from the risk factor simulation process; anddetermine an initial margin for the at least one financial portfoliobased on the portfolio profit and loss values. The instructions alsocause the system to execute the LRC model, causing the LRC model todetermine a portfolio level liquidity risk for the at least onefinancial portfolio, based on the additional data and portfolio profitand loss data from the margin model. The LRC model determines theliquidity risk based on one or more equivalent portfolio representationsof the at least one portfolio.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing summary and following detailed description may be betterunderstood when read in conjunction with the appended drawings.Exemplary embodiments are shown in the drawings, however, it should beunderstood that the exemplary embodiments are not limited to thespecific methods and instrumentalities depicted therein. In thedrawings:

FIG. 1 shows an exemplary risk engine architecture.

FIGS. 2, 2A and 2B, collectively “FIG. 2,” show an exemplary diagramshowing various data elements and functions of an exemplary systemaccording to the present disclosure.

FIG. 3 shows an exemplary implied volatility to delta surface graph ofan exemplary system according to the present disclosure.

FIG. 4 shows a cross-section of the exemplary implied volatility of anexemplary system according to the present disclosure.

FIG. 5 shows an exemplary implied volatility data flow of an exemplarysystem according to the present disclosure.

FIG. 6 shows a graphical representation of an exemplary transformationof delta-to-strike of an exemplary system according to the presentdisclosure.

FIG. 7 shows an exemplary fixed time series of an exemplary systemaccording to the present disclosure.

FIG. 8 shows a chart of the differences between an exemplary relativeand fixed expiry data series of an exemplary system according to thepresent disclosure.

FIG. 9 shows a chart of an exemplary fixed expiry dataset of anexemplary system according to the present disclosure.

FIG. 10 shows an exemplary clearinghouse account hierarchy of anexemplary system according to the present disclosure.

FIG. 11 shows another exemplary clearinghouse account hierarchy of anexemplary system according to the present disclosure.

FIG. 12 shows another exemplary clearinghouse account hierarchy of anexemplary system according to the present disclosure.

FIG. 13 shows another exemplary clearinghouse account hierarchy of anexemplary system according to the present disclosure.

FIG. 14 shows another exemplary clearinghouse account hierarchy of anexemplary system according to the present disclosure.

FIG. 15 shows an exemplary hierarchy of a customer's account portfolioof an exemplary system according to the present disclosure.

FIG. 16 is a functional block diagram of an example risk managementsystem according to the present disclosure.

FIG. 17 is a functional block diagram of an example margin model of therisk management system shown in FIG. 16 according to the presentdisclosure.

FIG. 18 is a functional block diagram of an example LRC model of therisk management system shown in FIG. 16 according to the presentdisclosure.

FIG. 19 is a flowchart diagram of an example method of determining riskexposure and adjust parameters of the model system shown in FIG. 16according to the present disclosure.

FIG. 20 is a flowchart diagram of an example method of determining atleast one initial margin according to the present disclosure.

FIG. 21 is a flowchart diagram of an example method of generating avolatility forecast according to the present disclosure.

FIG. 22 is a flowchart diagram illustrating an example method ofdetermining an initial margin based on a volatility forecast accordingto the present disclosure.

FIG. 23A is a functional block diagram of an example signal flow fordetermining the volatility forecast in connection with FIG. 21 accordingto the present disclosure.

FIG. 23B is a functional block diagram of an example signal flow fordetermining an initial margin in connection with FIG. 22 according tothe present disclosure.

FIG. 24 is a flowchart diagram illustrating an example method ofdetermining an LRC value for at least one portfolio according to thepresent disclosure.

FIG. 25 is a functional block diagram of an example signal flow fordetermining an LRC value by an LRC model according to the presentdisclosure.

FIG. 26 is an illustration of example grouping methods for determiningthe LRC value according to the present disclosure.

FIG. 27 is a flowchart diagram of an example method of generatingsynthetic data according to the present disclosure.

FIG. 28 is a functional block diagram of an example computer system,according to the present disclosure.

DETAILED DESCRIPTION Introduction

The present disclosure relates generally to systems and methods forefficiently and accurately collateralizing counterparty credit risk.Notably, the systems and methods described herein are effective for usein connection with all types of financial products (e.g., linear andnon-linear, complex), and with portfolios of financial products, whetherfully or partially diversified.

As indicated above, conventional systems utilize a linear analysisapproach for modeling risk of all types of financial products, includingthose financial products that are not themselves linear in nature.Moreover, conventional systems fail to consider diversification orcorrelations between financial products in a portfolio when determiningan initial margin (“IM”) for the entire portfolio. As will beappreciated, diversification and product correlations within a portfoliocan offset some of the overall risk of the portfolio, thereby reducingthe IM that needs to be collected.

The systems and methods described herein address the foregoingdeficiencies (as well as others) by providing new systems and methodsthat efficiently and accurately calculate IMs for both linear andnon-linear products, and that consider diversification and productcorrelations when determining IM for a portfolio of products.

In one aspect, the present disclosure relates to a novel multi-assetportfolio simulation system and method (also referred to herein as ICERisk Model (IRM) system or margin model system). IRM, in one embodiment,utilizes a unique technique for determining IM that includes (withoutlimitation) decomposing products (e.g., complex non-linear products)into their respective individual components, and then mathematicallymodeling the components to assess a risk of each component. For purposesof this disclosure, “decomposing” may be considered a mapping of aparticular financial product to the components or factors that drivethat product's profitability (or loss). This mapping may include, forexample, identifying those components or factors that drive a financialproduct's profitability (or loss). A “component” or “factor” (or “riskfactor”) may therefore refer to a value, rate, yield, underlying productor any other parameter or object that may affect, negatively orpositively, a financial product's profitability.

Once the components (or factors) are mathematically modeled, a secondmapping (in the reverse direction) may be executed in which thecomponents (or factors) are then reassembled. In the context of thisdisclosure, “reassembling” components of a financial product may beconsidered an aggregation of the results of the modeling proceduresummarized above.

After the components (or factors) of the financial product arereassembled, the entire product may be processed through a filteredhistorical simulation (FHS) process to determine an IM (or a ‘marginrate’) for the financial product.

For purposes of this disclosure, the term “product” or “financialproduct” should be broadly construed to comprise any type of financialinstrument including, without limitation, commodities, derivatives,shares, bonds, and currencies. Derivatives, for example, should also bebroadly construed to comprise (without limitation) any type of options,caps, floors, collars, structured debt obligations and deposits, swaps,futures, forwards, and various combinations thereof.

A similar approach may be taken for a portfolio of financial products(i.e., a financial portfolio). Indeed, a financial portfolio may bebroken down into its individual financial products, and the individualfinancial products may each be decomposed into their respectivecomponents (or factors). Each component (or factor) may then bemathematically modeled to determine a risk associated with eachcomponent (or factor), reassembled to its respective financial product,and the financial products may then be reassembled to form the financialportfolio. The entire portfolio may then be processed through an FHSprocess to determine an overall margin rate for the financial portfolioas a whole.

In addition, any correlations between the financial products orpertinent product hierarchy within the financial portfolio may beconsidered and taken into account to determine an IM (or a margin rate)for the financial portfolio. This may be accomplished, for example, byidentifying implicit and explicit relationships between all financialproducts in the financial portfolio, and then accounting (e.g.,offsetting risk) for the relationships where appropriate.

As will be evident from the foregoing, the present disclosure relates toa top-down approach for determining IM that determines and offsetsproduct risk where appropriate. As a result, the systems and methodsdescribed herein are able to provide a greater level of precision andaccuracy when determining IM. In addition, this top-down approachfacilitates the ability to compute an IM on a fully diversified level orat any desired percentage level.

Systems and methods of the present disclosure may include and/or beimplemented by one or more computers or computing devices. For purposesof this disclosure, a “computer” or “computing device” (these terms maybe used interchangeably) may be any programmable machine capable ofperforming arithmetic and/or logical operations. In some embodiments,computers may comprise processors, memories, data storage devices,and/or other commonly known or novel components. These components may beconnected physically or through network or wireless links. Computers mayalso comprise software which may direct the operations of theaforementioned components.

Exemplary (non-limiting) examples of computers include any type ofserver (e.g., network server), a processor, a microprocessor, a personalcomputer (PC) (e.g., a laptop computer), a palm PC, a desktop computer,a workstation computer, a tablet, a mainframe computer, an electronicwired or wireless communications device such as a telephone, a cellulartelephone, a personal digital assistant, a voice over Internet protocol(VOIP) phone or a smartphone, an interactive television (e.g., atelevision adapted to be connected to the Internet or an electronicdevice adapted for use with a television), an electronic pager or anyother computing and/or communication device.

Computers may be linked to one another via a network or networks and/orvia wired or wired communications link(s). A “network” may be anyplurality of completely or partially interconnected computers whereinsome or all of the computers are able to communicate with one another.The connections between computers may be wired in some cases (i.e. viawired TCP connection or other wired connection) or may be wireless (i.e.via WiFi network connection). Any connection through which at least twocomputers may exchange data can be the basis of a network. Furthermore,separate networks may be interconnected such that one or more computerswithin one network may communicate with one or more computers in anothernetwork. In such a case, the plurality of separate networks mayoptionally be considered to be a single network.

Terms and Concepts

The following terms and concepts may be used to better understand thefeatures and functions of systems and methods according to the presentdisclosure:

Account refers to a topmost level within a customer portfolio in themargin account hierarchy (discussed below) where a final margin isreported; the Account is made up of Sectors (discussed below).

Backfilling See Synthetic Price Service (defined below).

Backtesting refers to a normal statistical framework that consists ofverifying that actual losses are in line with projected losses. Thisinvolves systematically comparing the history of VaR (defined below)forecasts with their associated portfolio returns. Three exemplarybacktests may be used to measure the performance of margining accordingto the present disclosure: Basel Traffic Light, Kupiec, andChristofferson Tests.

Basel Traffic Light Test refers to a form of backtesting which tests ifthe margin model has too many margin breaches.

Bootstrapping See Correlation Matrix Joint Distribution (defined below).

Christoffersen Test refers to a form of backtesting which tests if themargin model has too many or too few margin breaches and whether themargin breaches were realized on consecutive days.

Cleaned Data See Synthetic Data (defined below).

Cleaned Historical Dynamic Data (CHDD) refers to a process to clean theraw time series data and store the processed financial time series datato be fed into a margin model as input.

Conditional Coverage relates to backtesting and takes into account thetime in which an exceptions occur. The Christoffersen test is an exampleof conditional coverage.

Confidence Interval defines the percentage of time that an entity (e.g.,exchange firm) should not lose more than the VaR amount.

Contingency Group (CG) refers to collections of products that havedirect pricing implications on one another; for instance, an option on afuture and the corresponding future. An example of a CG is Brent={B,BUL, BRZ, BRM, . . . }, i.e., everything that ultimately refers to Brentcrude as an underlying for derivative contracts.

Contract refers to any financial instrument (i.e., any financialproduct) which trades on a financial exchange and/or is cleared at aclearinghouse. A contract may have a PCC (physical commodity code), astrip date (which is closely related to expiry date), a pricing type(Futures, Daily, Avg., etc.), and so on.

Correlation Matrix Joint Distribution refers to a Synthetic PriceService (defined below) approach which builds a correlation matrix usingavailable time series on existing contracts which have sufficienthistorical data (e.g., 1,500 days). Once a user-defined correlationvalue is set between a target series (i.e., the product which needs tobe backfilled) and one of an existing series with sufficient historicaldata, synthetic returns for the target can be generated based on thecorrelation.

Coverage Ratio refers to a ratio comparing Risk Charge (defined below)to a portfolio value. This ration may be equal to the margin generatedfor the current risk charge day divided by the latest availableportfolio value.

DB Steering refers to an ability to manually or systematically setvalues in a pricing model without creating an offset between twopositions. This may be applicable to certain instruments that are notcorrelated or fully correlated both statistically and logically (e.g.,Sugar and Power).

Diversification Benefit (DB) refers to a theoretical reduction in risk afinancial portfolio achieved by increasing the breadth of exposures tomarket risks over the risk to a single exposure.

Diversification Benefit (DB) Coefficient refers to a number between 0and 1 that indicates the amount of diversification benefit allowed forthe customer to receive. Conceptually, a diversification benefitcoefficient of zero may correspond to the sum of the margins for thesub-portfolios, while a diversification benefit coefficient of 1 maycorrespond to the margin calculated on the full portfolio.

Diversification Benefit (DB) Haircut refers to the amount of thediversification benefit charged to a customer or user, representing areduction in diversification benefit.

Dynamic VaR refers to the VaR of a portfolio assuming that theportfolio's exposure is constant through time.

Empirical Characteristic Function Distribution Fitting (ECF) refers to abackfilling approach which fits a distribution to a series of returnsand calculates certain parameters (e.g., stability α, scale σ, skewnessβ, and location μ) in order to generate synthetic returns for any gapssuch that they fall within the same calculated distribution.

Enhanced Historical Simulation Portfolio Margining (EHSPM) refers to aVaR risk model which scales historical returns to reflect current marketvolatility using EWMA (defined below) for the volatility forecast. RiskCharges are aggregated according to Diversification Benefits.

Estimated Weighted Moving Average (EWMA) is used to place emphasis onmore recent events versus past events while remembering passed eventswith decreasing weight.

Exceedance may be referred to as margin breach in backtesting and may beidentified when Variation Margin is greater than a previous day'sInitial Margin.

Exponentially Weighted Moving Average (EWMA) refers to a model used totake a weighted average estimation of returns.

Filtered Data refers to option implied volatility surfaces truncated at(e.g., seven) delta points.

Fixed Expiry refers to a fixed contract expiration date. As timeprogresses, the contract will move closer to its expiry date (i.e., timeto maturity is decaying). For each historical day, settlement data whichshare the same contract expiration date may be obtained to form a timeseries, and then historical simulation may be performed on that series.

Haircut refers to a reduction in the diversification benefit,represented as a charge to a customer.

Haircut Contribution refers to a contribution to the diversificationhaircut for each pair at each level.

Haircut Weight refers to the percentage of the margin offsetcontribution that will be haircut at each level.

Historical VaR uses historical data of actual price movements todetermine the actual portfolio distribution.

Holding Period refers to a discretionary value representing the timehorizon analyzed, or length of time determined to be required to holdassets in a portfolio.

Implied Volatility Dynamics refers to a process to compute the scaledimplied volatilities using the Sticky-Delta or Sticky-Strike method(defined below). It may model the implied volatility curve as sevenpoints on the curve.

Incremental VaR refers to the change in Risk of a portfolio given asmall trade. This may be calculated by using the marginal VaR times thechange in position.

Independence In backtesting, Independence takes into account when anexceedance or breach occurs.

Initial Margin (IM) refers to an amount of collateral that a holder of aparticular financial product (or financial portfolio) must deposit tocover for default risk.

Input Data refers to Raw data that is filtered into cleaned financialtime series. The cleaned time series may be input into a historicalsimulation. New products or products without a sufficient length of timeseries data have proxy time series created.

Kupiec Test refers to a process for testing, in the context ofbacktesting, which tests if a margin model has too many or too fewmargin breaches.

Instrument Correlation refers to a gain in one instrument that offsets aloss in another instrument on a given day. At a portfolio level, for Xdays of history (e.g.,), a daily profit and loss may be calculated andthen ranked.

Margin Attribution Report defines how much of a customer's initialmargin charge was from active trading versus changes in the market. In aportfolio VaR model, one implication is that customers' initial margincalculation will not be a sub-process of the VaR calculation.

Margin Offset Contribution refers to the diversification benefit of afinancial products to a portfolio (e.g., the offset contribution ofcombining certain financial products into the same portfolio versusmargining the financial products separately).

Margin Testing—Risk Charge testing may be done to assess how a riskmodel performs on a given portfolio or financial product. Theperformance tests may be run on-demand and/or as a separate process,distinct from the production initial margin process. Backtesting may bedone on a daily, weekly, or monthly interval (or over any period).Statistical and regulatory tests may be performed on each modelbacktest. Margin Tests include (without limit) the Basel Traffic light,Kupiec, and Christofferson test.

Marginal VaR refers to the proportion of the total risk to each RiskFactor. This provides information about the relative risk contributionfrom different factors to the systematic risk. The sum of the marginalVaRs is equal to the systematic VaR.

Offset refers to a decrease in margin due to portfolio diversificationbenefits.

Offset Ratio refers to a ratio of total portfolio diversificationbenefit to the sum of pairwise diversification benefits. This ratioforces the total haircut to be no greater than the sum of offsets ateach level so that the customer is never charged more than the offset.

Option Pricing refers to options that are repriced using the scaledunderlying and implied volatility data.

Option Pricing Library—Since underlying prices and option impliedvolatilities are scaled separately in the IRM option risk chargecalculation process, an option pricing library may be utilized tocalculate the option prices from scaled underlying prices and impliedvolatilities. The sticky Delta technique may also utilize conversionsbetween option strike and delta, which may be achieved within the optionpricing library.

Overnight Index Swap (OIS) refers to an interest rate swap involving anovernight rate being exchanged for a fixed interest rate. An overnightindex swap uses an overnight rate index, such as the Federal Funds Rate,for example, as the underlying for its floating leg, while the fixed legwould be set at an assumed rate.

Portfolio Bucketing refers to a grouping of clearing member's portfolios(or dividing clearing member's account) in a certain way such that therisk exposure of the clearinghouse can be evaluated at a finer grain.Portfolios are represented as a hierarchy from the clearing member tothe instrument level. Portfolio bucketing may be configurable to handlemultiple hierarchies.

Portfolio Compression refers to a process of mapping a portfolio to aneconomically identical portfolio with a minimal set of positions. Theprocess of portfolio compression only includes simple arithmetic tosimplify the set of positions in a portfolio.

Portfolio Risk Aggregation refers to the aggregated risk charge for eachportfolio level from bottom-up.

Portfolio Risk Attribution refers to the risk attribution for eachportfolio from top-down.

Portfolio VaR refers to a confidence on a portfolio, where VaR is a riskmeasure for portfolios. As an example, VaR at a ninety-nine percent(99%) level may be used as the basis for margins.

Position In the Margin Account Hierarchy (discussed below), the positionlevel is made up of distinct positions in the cleared contracts within acustomer's account. Non-limiting examples of positions may include 100lots in Brent Futures, −50 lots in Options on WTI futures, and −2,500lots in AECO Basis Swaps.

Product as indicated above, a product (or financial product) may referto any financial instrument. In fact, the terms product and instrumentmay be used interchangeably herein. In the context of a Margin AccountHierarchy, Products may refer to groups of physical or financial claimson a same (physical or financial) underlying. Non-limiting examples ofProducts in this context may include Brent Futures, Options on WTIfutures, AECO Natural Gas Basis swaps, etc.

Raw Data refers to data which is obtained purely from trading activityrecorded via a settlement process.

Raw Historical Dynamic Data (RHDD) refers to an ability to storehistorical financial time series for each unique identifier in thestatic data tables for each historical day (e.g., expiration date,underlying, price, implied volatility, moneyness, option Greeks, etc.).

Relative Expiry—As time progresses, a contract remains at the samedistance to its expiry date and every point in the time seriescorresponds to different expiration dates. For each historical day,settlement data which share the same time to maturity may be used toform the time series.

Reporting refers to the reporting of margin and performance analytics ateach portfolio hierarchy. A non-limiting example of a portfoliohierarchy grouping includes: Clearing Member, Clearing Member Client,Product type, Commodity type, instrument. Backtest reporting may beperformed on regular intervals and/or on-demand.

Return Scaling refers to a process to compute and scale returns for eachunderlying instrument and implied volatility in the CHDD. Scaling may bedone once settlement prices are in a clearing system.

Risk Aggregation refers to a process to aggregate risk charges from asub-portfolio level to a portfolio level. This aggregation may beperformed using the diversification benefits to off-set.

Risk Attribution refers to a process to attribute contributions to therisk charges of portfolios to sub-portfolios.

Risk Charge refers to an Initial Margin applied to on the risk chargedate.

Risk Charge Performance Measurement refer to the performance metricsthat are calculated on each backtest, which can be performed atspecified intervals and/or by request.

Risk Dashboard refers to a risk aggregation reporting tool (optionallyimplemented in a computing device and accessible via a Graphical UserInterface (GUI)) for risk charges across all portfolio hierarchies. TheRisk Dashboard may be configured to provides the ability to drill downinto detailed analysis and reports across the portfolio hierarchies.

Risk Factors—As indicated above, a Risk Factor may refer to any value,rate, yield, underlying product or any other parameter or object thatmay affect, negatively or positively, a financial product'sprofitability. Linear instruments may themselves be a risk factor. Foreach option product, the underlying instrument for every option expirymay be a risk factor. Seven (7) points on the implied volatility curvefor every option expiry may also be risk factors.

Sector refers to a level of the Margin Account Hierarchy containingcontingency groups. Non-limiting examples of sectors include NorthAmerican Power, North American Natural Gas, UK Natural Gas, EuropeanEmissions, etc.

Specific VaR refers to the Risk that is not captured by mapping aportfolio to risk factors.

Static VaR refers to the VaR of a portfolio assuming that theportfolio's positions are constant through time.

Sticky Delta Rule refers to a rule formulated under the assumption thatimplied volatility tends to “stick” to delta. The sticky delta rule maybe used by quoting implied volatility with respect to delta. Havinginput a set of fixed deltas, for example, historical impliedvolatilities which come from pairing each delta to a unique option andmatches each input delta with the option whose delta is closest to thisinput value may be obtained. This process results in an impliedvolatility surface.

Synthetic Data corresponds to any data which has required SyntheticPrice Service to backfill prices or fill in gaps where data is lacking.

Synthetic Price Service, also referred to as Backfilling, refers to aprocess to logically simulate historical price data where it did notexist, with the goal of building a historical profit and loss simulationto submit into a VaR (Value at Risk) calculation. Non-limiting exemplaryalgorithms that may be utilized to generate synthetic prices include(without limitation): Empirical Characteristic Function DistributionFitting (ECF) and Correlation Matrix Joint Distribution (e.g.,Bootstrapping).

Systematic VaR refers to the Risk that is captured by mapping aportfolio to risk factors.

Time Series corresponds to any data which has required Synthetic PriceService to backfill prices or fill in gaps where data is lacking.

Total VaR refers to Systematic VaR plus Specific VaR.

Unconditional Coverage—In backtesting, these tests statistically examinethe frequency of exceptions over some time interval. Basel Traffic Lightand Kupiec can both be classified as non-limiting examples ofunconditional coverage tests.

VaR (Value at Risk) refers to the maximum loss a portfolio is expectedto incur over a particular time period with a specified probability.

Variation Margin (VM) refers to margin paid on a daily or intraday basisbased on adverse price movements in contracts currently held in anaccount. VM may be computed based on the difference between dailysettlement prices and the value of the instrument in a given portfolio.

Volatility Cap or Volatility Ceiling refers to an upper limit on howhigh a current backtesting day's forecasted volatility is allowed tofluctuate with respect to a previous backtesting day. The Volatility Capmay be implemented by using a multiplier which defines this upper limit.A Volatility Cap may be used to prevent a system from posting a veryhigh margin requirement due to a spike in market volatility.

Volatility Forecast refers to risk factor return volatility that isforecasted using an EWMA. The EWMA model may weight recent informationmore than past information which makes the risk factor return volatilitymore adaptive than a standard volatility estimate.

Yield Curve describes interest rates (cost of borrowing) plotted againsttime to maturity (term of borrowing) and is essential to pricingoptions.

Yield Curve Generator (YCG) refers to an algorithm which produces fullYield Curves by interpolating/extrapolating Overnight Index Swap (OIS)rates.

Overview

As noted above, the systems and methods of this disclosure provide amodel for more efficiently and accurately determining initial margin.This new model (among other things) is able to scale linearly with thenumber of underlyings so that the introduction of new products or assetclasses does not require an outsized amount of human interaction andongoing maintenance. The model also allows control of diversificationbenefits at multiple levels in order to maintain a conservative bias,and may be explainable without large amounts of complex mathematics.

The present disclosure takes an empirical approach to the risk ofportfolios of financial products. As further discussed below, historicalsimulation may be utilized (as part of the margin model) to minimize theamount of prescription embedded within the risk charge framework, whichallows for a more phenomenological approach to risk pricing that tiesthe results back to realized market events. The aim has been to make theframework as simple as possible while retaining the core functionalityneeded.

Features of the model include (without limitation): utilizing a VaR asthe risk measure; determining initial margin based on historical return;scaling market volatility of historical returns to reflect currentmarket volatility; scaling each product in isolation and withoutconsidering the market volatility of all other assets; volatilityforecasting based on EWMA; full revaluation across the historical periodfor every position; sticky delta evolution of an option impliedvolatility surface; modeling an implied volatility surface using deltapoints (e.g., seven points) on a curve; dynamic VaR over holdingperiods; aggregating risk charges according to diversification benefits;calculating diversification benefits (DBs) from historical data (DBs canbe prescribed as well); performance analysis on sufficient capitalcoverage and model accuracy; as well as others that will be apparentbased on the following descriptions.

The systems and methods of this disclosure may apply to any type offinancial products and combinations thereof, including (withoutlimitation): futures, forwards, swaps, ‘vanilla’ options (calls andputs), basic exercise (European and American), options (includingoptions on first line swaps), fixed income products (e.g., swaps (IRS,CDS, Caps, Floors, Swaptions, Forward Starting, etc.)), dividendpayments, exotic options (e.g., Asian Options, Barrier Options,Binaries, Lookbacks, etc.), exercise products (e.g., Bermudan, Canary,Shout, Swing, etc.).

The model of the present disclosure may, in an exemplary embodiment,operate under the following assumptions, although said model may beimplemented under additional, alternative or fewer assumptions:

a. future volatility of financial returns may be estimated from the pastvolatility of financial returns;

b. future (forecasts) may be similar to past performance (e.g.,volatility, correlations, credit events, stock splits, dividendpayments, etc.);

c. EWMA may be utilized to estimate return volatility;

d. an EWMA decay factor (e.g., of 0.97) may be used to weight historicalreturns;

e. volatility scaling historical returns data to resemble more recentreturn volatility may be utilized to forecast future return volatility;

f. the volatility of individual underlying products may be adjustedindividually;

g. portfolio exposures may be assumed constant over a holding period;

h. the model assumes accurate data is input;

i. disparity in local settlement time does not adversely impact theaccuracy of the volatility forecast;

j. a 99% VaR for a 1,000 day return series can be accurately estimated;

k. option implied volatility surface dynamics are relative to thecurrent underlying instrument's price level; and

l. full position valuation may be performed across historical windows of1,000 days or more.

Types of information and data that may be utilized by the model mayinclude (without limitation): financial instrument data (e.g., staticdata (instrument properties), dynamic data (prices, impliedvolatilities, etc.)), portfolios (composition, diversification benefits,etc.), risk model configurations (e.g., EWMA decay factor, VaR level,days of historical returns, etc.).

Components of a risk information system according to the presentdisclosure may include (without limitation): a financial instrumentdatabase (to store instrument properties, historical data, etc.), a datafilter (to clean erroneous data, fill gaps in data, convert raw datainto a time series, etc.), portfolio bucketing (to group portfolios byclearing member, client accounts, product, commodity, market type,etc.), portfolio compression (to net portfolios to a minimal set ofpositions, e.g., currency triangles, long and shorts on the sameinstrument, etc.), financial pricing library (e.g., option pricing,implied volatility dynamics, returns calculations, return scaling,etc.), currency conversion (e.g., converts returns to a common returncurrency for portfolios that contain positions in instruments with morethan one settlement currency), risk library (to compute risk at theinstrument level, compute risk at the portfolio levels, applydiversification benefits, etc.), performance analysis library (toperform backtests, compute performance measures, produce summary reportsand analytics, etc.).

Turning now to FIG. 1, an exemplary risk engine architecture 100 isshown. This exemplary architecture 100 includes a risk engine managerlibrary 101 that provides main functionality for the architecture 100and a communication library 102 that provides data communicationfunctionality. Components such as a risk server 105 and a cluster of oneor more servers 106 may provide data and information to thecommunication library 102. Data and information from the communicationlibrary 102 may be provided to a risk engine interface library 103,which provides an ‘entrance’ (e.g., daily risk calculation entrance andbacktesting functionality entrance) into the risk engine calculationlibrary 104. The risk engine calculation library 104 may be configuredto perform daily risk calculations and backtesting functions, as well asall sub-functions associated therewith (e.g., data cleaning, time seriescalculations, option calculations, etc.).

The exemplary architecture 100 also may include a unit test library 107,in communication with the communication library 102, risk engineinterface library 103 and risk engine calculation library 104, toprovide unit test functions. A utility library 108 may be provided incommunication with both the risk engine interface library 103 and therisk engine calculation library 104 to provide in/out (I/O) functions,conversion functions and math functions.

A financial engineering library 109 may be in communication with theutility library 108 and the risk engine calculation library 104 toprovide operations via modules such as an option module, time seriesmodule, risk module, simulation module, analysis module, etc.

A reporting library 110 may be provided to receive data and informationfrom the risk engine calculation library 104 and to communicate with theutility library 108 to provide reporting functions.

Notably, the various libraries, modules and functions described above inconnection with the exemplary architecture 100 of FIG. 1 may comprisesoftware components (e.g. computer-readable instructions) embodied onone or more computing devices (co-located or across various locations,in communication via wired and/or wireless communications links), wheresaid computer-readable instructions are executed by one or moreprocessing devices to achieve and provide their respective functions.

Turning now to FIGS. 2, 2A and 2B, collectively referred to as “FIG. 2”hereafter, an exemplary diagram 200 showing the various data elementsand functions of an exemplary IRM system according to the presentdisclosure is shown. More particularly, the diagram 200 shows the dataelements and functions provided in connection with products 201, prices202, returns 203, market risk adaptation 204, historical simulation 205,portfolios 206, margins 207 and reporting 208, and their respectiveinteractions. These data components and functions may be provided inconnection with (e.g., the components may be embodied on) systemelements such as databases, processors, computer-readable instructions,computing devices (e.g., servers) and the like.

An exemplary computer-implemented method of collateralizing counterpartycredit risk in connection with one or more financial products mayinclude receiving as input, by at least one computing device, datadefining at least one financial product. The computing device mayinclude one or more co-located computers, computers dispersed acrossvarious locations, and/or computers connected (e.g., in communicationwith one another) via a wired and/or wireless communications link(s). Atleast one of the computing devices comprises memory and at least oneprocessor executing computer-readable instructions to perform thevarious steps described herein.

Upon receiving the financial product data, the exemplary method mayinclude mapping, by computing device(s), the financial product(s) to atleast one risk factor, where this mapping step may include identifyingat least one risk factor that affects a profitability of the financialproduct(s).

Next, the method may include executing, by the computing device(s), arisk factor simulation process involving risk factor(s) previouslyidentified. This risk factor simulation process may include retrieving,from a data source, historical pricing data for the one risk factor(s),determining statistical properties of the historical pricing data,identifying any co-dependencies between prices that exist within thehistorical pricing data and generating, as output, normalized historicalpricing data based on the statistical properties and co-dependencies.

The risk factor simulation process may also include a filteredhistorical simulation process, which may itself include a co-variancescaled filtered historical simulation that involves normalizing thehistorical pricing data to resemble current market volatility byapplying a scaling factor to said historical pricing data. This scalingfactor may reflect the statistical properties and co-dependencies of thehistorical pricing data.

Following the risk factor simulation process, the exemplary method mayinclude generating, by the computing device(s), product profit and lossvalues for the financial product(s) based on output from the risk factorsimulation process. These profit and loss values may be generated bycalculating, via a pricing model embodied in the computing device(s),one or more forecasted prices for the financial product(s) based on thenormalized historical pricing data input into the pricing model, andcomparing each of the forecasted prices to a current settlement price ofthe financial product(s) to determine a product profit or loss valueassociated with each of said forecasted prices.

Next, the computing device(s) may determine an initial margin for thefinancial product(s) based on the product profit and loss values, whichmay include sorting the product profit and loss values, most profitableto least profitable or vice versa and selecting the product profit orloss value among the sorted values according to a predeterminedconfidence level, where the selected product profit or loss valuerepresents said initial margin.

In one exemplary embodiment, the historical pricing data may includepricing data for each risk factor over a period of at least one-thousand(1,000) days. In this case, the foregoing method may involve:calculating, via the pricing model, one-thousand forecasted prices, eachbased on the normalized pricing data pertaining to a respective one ofthe one-thousand days; determining a product profit or loss valueassociated with each of the one-thousand forecasted prices by comparingeach of the one-thousand forecasted prices to a current settlement priceof the at least one financial product; sorting the product profit andloss values associated with each of the one-thousand forecasted pricesfrom most profitable to least profitable or vice versa; and identifyinga tenth least profitable product profit or loss value. This tenth leastprofitable product profit or loss value may represent the initial marginat a ninety-nine percent confidence level.

An exemplary computer-implemented method of collateralizing counterpartycredit risk in connection with a financial portfolio may includereceiving as input, by one or more computing device(s), data defining atleast one financial portfolio. The financial portfolio(s) may itselfinclude one or more financial product(s). As with the exemplary methoddiscussed above, the computing device(s) used to implement thisexemplary method may include one or more co-located computers, computersdispersed across various locations, and/or computers connected (e.g., incommunication with one another) via a wired and/or wirelesscommunications link(s). At least one of the computing devices comprisesmemory and at least one processor executing computer-readableinstructions to perform the various steps described herein.

Upon receiving the financial portfolio data, the exemplary method mayinclude mapping, by the computing device(s), at least one financialproduct in the portfolio to at least one risk factor by identifying atleast one risk factor that affects a probability of said financialproduct(s).

Next, the computing device(s) may execute a risk factor simulationprocess involving the risk factor(s). This risk factor simulationprocess may include retrieving, from a data source, historical pricingdata for the risk factor(s) and determining statistical properties ofthe historical pricing data. Then, any co-dependencies between pricesthat exist within the historical pricing data may be identified, and anormalized historical pricing data may be generated based on thestatistical properties and the co-dependencies.

The risk factor simulation process may further include a filteredhistorical simulation process. This filtered historical simulationprocess may include a co-variance scaled filtered historical simulationthat involves normalizing the historical pricing data to resemblecurrent market volatility by applying a scaling factor to the historicaldata. This scaling factor may reflect the statistical properties andco-dependencies of the historical pricing data.

Following the risk factor simulation process, the exemplary method mayinclude generating, by the computing device(s), product profit and lossvalues for the financial product(s) based on output from the risk factorsimulation process. Generating these profit and loss values may includecalculating, via a pricing model embodied in the computing device(s),one or more forecasted prices for the financial product(s) based on thenormalized historical pricing data input into said pricing model; andcomparing each of the forecasted prices to a current settlement price ofthe at financial product(s) to determine a product profit or loss valueassociated with each of said forecasted prices.

The profit and loss values of the respective product(s) may then beaggregated to generate profit and loss values for the overall financialportfolio(s). These portfolio profit and loss values may then be used todetermine an initial margin for the financial portfolio(s). In oneembodiment, the initial margin determination may include sorting theportfolio profit and loss values, most profitable to least profitable orvice versa; and then selecting the portfolio profit or loss value amongthe sorted values according to a predetermined confidence level. Theselected portfolio profit or loss value may represent the initialmargin.

In one exemplary embodiment, the historical pricing data may includepricing data for each risk factor over a period of at least one-thousand(1,000) days and the financial portfolio may include a plurality offinancial products. In this case, the foregoing method may involve:calculating, via the pricing model, one-thousand forecasted prices foreach of the plurality of financial products, where the forecasted pricesare each based on the normalized pricing data pertaining to a respectiveone of the one-thousand days; determining one-thousand product profit orloss values for each of the plurality of financial products by comparingthe forecasted prices associated each of the plurality of financialproducts to a respective current settlement price; determiningone-thousand portfolio profit or loss values by aggregating a respectiveone of the one-thousand product profit or loss values from each of theplurality of financial products; sorting the portfolio profit and lossvalues from most profitable to least profitable or vice versa; andidentifying a tenth least profitable portfolio profit or loss value.This tenth least profitable product profit or loss value may representthe initial margin at a ninety-nine percent confidence level.

An exemplary system configured for collateralizing counterparty creditrisk in connection with one or more financial products and/or one ormore financial portfolios may include one or more computing devicescomprising one or more co-located computers, computers dispersed acrossvarious locations, and/or computers connected (e.g., in communicationwith one another) via a wired and/or wireless communications link(s). Atleast one of the computing devices comprises memory and at least oneprocessor executing computer-readable instructions that cause theexemplary system to perform one or more of various steps describedherein. For example, a system according to this disclosure may beconfigured to receive as input data defining at least one financialproduct; map the financial product(s) to at least one risk factor;execute a risk factor simulation process (and/or a filtered historicalsimulation process) involving the risk factor(s); generate productprofit and loss values for the financial product(s) based on output fromthe risk factor simulation process; and determine an initial margin forthe financial product(s) based on the product profit and loss values.

Another exemplary system according to this disclosure may include atleast one computing device executing instructions that cause the systemto receive as input data defining at least one financial portfolio thatincludes at least one financial product; map the financial product(s) toat least one risk factor; execute a risk factor simulation process(and/or a filtered historical simulation process) involving the riskfactor(s); generate product profit and loss values for the financialproduct(s) based on output from the risk factor simulation process;generate portfolio profit and loss values for the financial portfoliobased on the product profit and loss values; and determine an initialmargin for the financial portfolio(s) based on the portfolio profit andloss values.

A more detailed description of features and aspects of the presentdisclosure are provided below.

Volatility Forecasting

A process for calculating forecasted prices may be referred to asvolatility forecasting. This process involves creating “N” number ofscenarios (generally set to 1,000 or any other desired number)corresponding to each risk factor of a financial product. The scenariosmay be based on historical pricing data such that each scenario reflectspricing data of a particular day. For products such as futurescontracts, for example, a risk factor for which scenarios may be createdmay include the volatility of the futures' price; and for options,underlying price volatility and the option's implied volatility may berisk factors. As indicated above, interest rate may be a further riskfactor for which volatility forecasting scenarios may be created.

The result of this volatility forecasting process is to create N numberof scenarios, or N forecasted prices, indicative of what could happen inthe future based on historical pricing data, and then calculate thedollar value of a financial product or of a financial portfolio (basedon a calculated dollar value for each product in the portfolio) based onthe forecasted prices. The calculated dollar values (of a product or ofa financial portfolio) can be arranged (e.g., best to worst or viceversa) to select the fifth percentile worst case scenario as theValue-at-Risk (VaR) number. Note here that any percentile can be chosen,including percentiles other than the first through fifth percentiles,for calculating risk. This VaR number may then be used to determine aninitial margin (IM) for a product or financial portfolio.

In one embodiment, the methodology used to perform volatilityforecasting as summarized above may be referred to as an “exponentiallyweighted moving average” or “EMWA” methodology. Inputs into thismethodology may include a scaling factor (λ) that may be set by aprogrammed computer device and/or set by user Analyst, and price seriesdata over “N” historical days (prior to a present day). For certainfinancial products (e.g., options), the input may also include impliedvolatility data corresponding to a number of delta points (e.g., seven)for each of the “N” historical days and underlying price data for eachof the “N” historical days.

Outputs of this EMWA methodology may include a new simulated series ofrisk factors, using equations mentioned below.

For certain financial products such as futures, for example, the EMWAmethodology may include:

-   -   1. Determining fix parameter values (N):

$\begin{matrix}{{N = {1000}},{\lambda = {{.9}7}}} & (1)\end{matrix}$

-   -   2. Gathering instrument price series (F_(t)):        -   F_(t), F₁₀₀₀, F₉₉₉, . . . F₁, where F₁₀₀₀ is a current day's            settlement price    -   3. Calculating Log returns r_(i):

$\begin{matrix}{r_{i} = {\log\frac{F_{i}}{F_{i - 1}}}} & (2)\end{matrix}$

-   -   4. Calculating sample mean of returns û:

$\begin{matrix}{\hat{u}{= {\frac{1}{N - 1}{\sum\limits_{i = 1}^{N - 1}r_{t}}}}} & (3)\end{matrix}$

-   -   5. Calculating sample variance of returns {circumflex over (v)}:

$\begin{matrix}{\overset{\hat{}}{v} = {\frac{1}{N - 2}{\sum\limits_{i = 1}^{N - 1}\left( {r_{i} - \overset{\hat{}}{u}} \right)^{2}}}} & (4)\end{matrix}$

-   -   6. Calculating EMWA scaled variance (ê_(j)), this may be the        first step of generating a volatility forecast: A first        iteration equation may use {circumflex over (v)}:

$\begin{matrix}{{\hat{e}}_{j} = {{\left( {1 - \lambda} \right)*\left( {r_{j} - \hat{u}} \right)} + {\lambda*\hat{v}}}} & (5)\end{matrix}$

-   -    then, a next iteration may proceed as:

$\begin{matrix}{{{\hat{e}}_{j} = {{\left( {1 - \lambda} \right)*\left( {r_{j} - \overset{\hat{}}{u}} \right)} + {\lambda*{\overset{\hat{}}{e}}_{j - 1}}}},} & (6)\end{matrix}$

-   -   -   where ê_(j-1) refers to value from previous iteration

    -   7. Calculating EMWA standardized log returns Z_(j):

$\begin{matrix}{{\overset{\hat{}}{z}}_{j} = {{\frac{\left( {r_{j} - {\overset{\hat{}}{u}}_{j}} \right)}{\sqrt{{\overset{\hat{}}{e}}_{j}}}\mspace{14mu}{or}\mspace{14mu}{\overset{\hat{}}{z}}_{j}} = \frac{r_{j}}{\sqrt{{\overset{\hat{}}{e}}_{j}}}}} & (7)\end{matrix}$

-   -   8. Calculating Volatility {circumflex over (σ)}_(J):

$\begin{matrix}{{\hat{\sigma}}_{j} = \sqrt{\max\left( {{\hat{v}}_{j},{\overset{\hat{}}{e}}_{j}} \right)}} & (8)\end{matrix}$

For other financial products, such as options for example, the EMWAmethodology may include performing all of the steps discussed above inthe context of futures (i.e., steps 1-8) for each underlying futureprice series and for the implied volatility pricing data correspondingto the delta points.

Implied Volatility Dynamics

When modeling risk for options, the “sticky delta rule” may be used inorder to accurately forecast option implied volatility. The ‘delta’ inthe sticky delta rule may refer to a sensitivity of an option's value tochanges in its underlying's price. Thus, a risk model system or methodaccording to this disclosure is able to pull implied volatilities forvanilla options and implied correlations for cal spread options (CSOs),for example, by tracking changes in option implied volatility in termsof delta.

More particularly, the sticky delta rule may be utilized by quotingimplied volatility with respect to delta. Having input a set of fixeddeltas, historical implied volatilities which come from pairing eachdelta to a unique option may be obtained. Each input delta may then bematched with the option whose delta is closest to this input value. Theimplied volatility for each of these options can then be associated witha fixed delta and for every day in history this process is repeated.Ultimately, this process builds an implied volatility surface using theimplied volatility of these option-delta pairs. An exemplary impliedvolatility to delta surface 300 is shown in FIG. 3.

Using an implied volatility surface, the implied volatility of anyrespective option may be estimated. In particular, systems and methodsaccording to this disclosure may perform a transformation from deltaspace to strike space for vanilla options in order to obtain a givenoption's implied volatility with respect to strike; for CSOs, strikesmay be pulled as well. In other words, given any strike, the systems andmethod of this disclosure can obtain its implied volatility.

The sticky delta rule is formulated under the impression that impliedvolatility tends to “stick” to delta. Under this assumption, changes inimplied volatility may be captured by tracking these “sticky deltas.”The present disclosure uses these “sticky deltas” as anchors in impliedvolatility surfaces which are then transformed to strike space in orderto quote a given option implied volatility.

Given inputs of implied volatilities of the “sticky deltas,” impliedvolatility for any given option may be determined. For CSOs, forexample, the delta to strike transformation may not be required, sinceimplied correlation is used to estimate prices.

A delta surface may be constructed using fixed delta points (e.g., sevenfixed delta points) and corresponding implied volatilities. Linearinterpolation may be used to find the implied volatility of a deltabetween any two fixed deltas. In practice, the implied volatilitysurface may be interpolated after transforming from delta space tostrike space. This way, the implied volatility for any strike may beobtained. A cross-section 400 of the exemplary implied volatilitysurface of FIG. 3 is shown in FIG. 4.

FIG. 5 shows an exemplary implied volatility data flow 500, whichillustrates how the EWMA scaling process 505 may utilize as few as one(or more) implied volatility 503 and one (or more) underlying price 504to operate. This is the case, at least in part, because (historical)implied volatility returns 501 and underlying price series returns 502are also inputs into the EWMA scaling process 505. EWMA scaling 505 isable to make these return series comparable in terms of a single inputprice 504 and a single implied volatility 503, respectively. In effect,EWMA scaling provides normalized or adapted implied volatilities 506 andunderlying prices 507.

The adapted implied volatilities 506 and underlying prices 507 may thenbe used by a Sticky Delta transformation process 508 to yield adaptedimplied volatilities with respect to Strike 509. This may then be fedinto an interpolation of surface process 510 to yield impliedvolatilities 511. The implied volatilities 511 as well as EWMA adaptedunderlying prices 507 may be utilized by an Option Pricer 512, togetherwith option parameters 513 to yield an EWMA adapted option series 514.

Transformation of Delta to Strike

In order to find the implied volatility for any given vanilla option orCSO, the systems and methods of the present disclosure may utilize atransformation of delta space to strike space. A graphicalrepresentation of an exemplary transformation of delta-to-strike 600 isshown in FIG. 6.

Given any strike, the present disclosure provides means for identifyingthe respective implied volatility. This transformation may be carriedout using the following formula, the parameters of which are defined inTable 1 below:

$\begin{matrix}{K = \frac{F}{\exp\left( {{\sigma{\sqrt{t} \cdot {N^{- 1}\left( {\exp\left( {rt} \right)} \right)}}} - {\frac{\sigma^{2}}{2}t}} \right)}} & (9)\end{matrix}$

TABLE 1 Delta to Strike Conversion Parameters Parameters DescriptionsN⁻¹(·) The inverse of the cumulative distribution of the standard normaldistribution F EWMA adapted price of the underlying future K Strikeprice σ Volatility of option returns t Time to expiry r Risk-free rateSystems and methods according to this disclosure may utilize impliedvolatility along with EWMA adapted forward price to estimate the priceof a financial product such as an option, for example. Details forcalculating the EWMA adapted forward price are discussed further below.

To capture the risk of options (although this process may apply to othertypes of financial instruments), for example, systems and methodsaccording to this disclosure may track risk factors associated with thefinancial product. In this example, the risk factors may include: anoption's underlying price and the option's implied volatility. As aninitial step, an implied volatility surface in terms of delta may becalculated. With this volatility surface, and using the sticky deltarule, the current level of implied volatility for any respective optionmay be determined.

Inputs for using the sticky delta rule may include: historicalunderlying prices, fixed deltas [if seven deltas are used, for example,they may include: 0.25, 0.325, 0.4, 0.5, 0.6, 0.675, 0.75], historicalimplied volatilities for each fixed delta, for CSOs, historical impliedcorrelations for each fixed delta, and for CSOs, historical strike foreach fixed delta.

Notably, when calculating VaR for options, implied volatilities may beused to estimate option price. Implied volatility in the options marketseems to move with delta. Using the sticky delta rule to track changesin implied volatility may therefore lead to accurate forecasts ofimplied volatility for all respective securities.

Volatility Ceiling

A volatility ceiling, or volatility cap, may be an upper limit on howhigh a current backtesting day's forecasted volatility is allowed tofluctuate with respect to a previous backtesting day. This volatilityceiling may be implemented by using a multiplier which defines thisupper limit. In a real-time system, which is forecasting margins insteadof using backtesting days, the terminology “yesterday's forecastedvolatility” may be used.

The idea of a volatility ceiling is to prevent the system from posting avery high margin requirement from the client due to a spike in marketvolatility. A margin call which requires the client to post a largemargin, especially during a market event, can to add to systemic risk(e.g., by ultimately bankrupting the client). Hence, the idea would beto charge a margin which is reasonable and mitigates clearinghouse risk.

If the volatility forecast for a future time period (e.g., tomorrow) isunreasonably high due to a volatility spike caused by a current day'srealized volatility, then it is possible that an unconstrained systemwould charge a very high margin to a client's portfolio underconsideration. Typically, this may occur when a market event hasoccurred related to the products in the client's portfolio. This canalso happen if there are ‘bad’ data points; typically, post backfilling,if returns generated fluctuate too much then this case can beencountered.

As noted above, charging a very high margin in case of a market eventcan add on to the systemic risk problem of generating more counterpartyrisk by potentially bankrupting a client that is already stretched oncredit. Hence, the present disclosure provides means for capping thevolatility and charging a reasonable margin which protects theclearinghouse and does not add to the systemic risk issue.

Inputs into a system for preventing an unreasonably high margin call mayinclude: a configurable multiplier alpha a (e.g., set to value 2),previous backtesting day's (or for live system yesterday's) forecastedvolatility, σ_(i-1) and current day's (e.g., today's) forecastedvolatility, σ_(i). Output of such a system may be based on followingequation:

$\begin{matrix}{{\sigma_{i} = {\min\left( {\sigma_{i},{\alpha*\sigma_{i - 1}}} \right)}},} & (10)\end{matrix}$

where σ_(i) is reassigned to a new volatility forecast, which is theminimum of today's volatility forecast, or alpha times yesterday'sforecast.

An initial step in the process includes defining a configurableparameter, alpha, which may be input directly into the system (e.g., viaa graphical user interface (GUI) embodied in a computing device incommunication with the system) and/or accepted from a control file.Then, the following steps can be followed for different types offinancial products.

For futures (or similar types of products):

1. For backtesting, a variable which holds previous backtesting day'svolatility forecast may be maintained in the system; and for a livesystem, yesterday's volatility forecast may be obtained in response to aquery of a database storing such information, for example.

2. The new volatility may be determined based on the following equation:

$\begin{matrix}{\sigma_{i} = {\min\left( {\sigma_{i},{\alpha*\sigma_{i - 1}}} \right)}} & (11)\end{matrix}$

For options (or similar types of products):

1. For backtesting, a vector of x-number (e.g., seven (7)) volatilityvalues for previous day corresponding to the same number (e.g., seven(7)) on delta points on a volatility surface may be maintained; and forthe live system, yesterday's volatility forecast for each of the sevendelta points on the volatility surface may be obtained in response to aquery of a database, for example.

2. The new volatility corresponding to each point may be determinedbased on the following equation:

$\begin{matrix}{{\sigma_{i}^{p} = {\min\left( {\sigma_{i}^{p},{\alpha*\sigma_{i - 1}^{p}}} \right)}},} & (12)\end{matrix}$

where p is delta point index

Under normal market conditions, a volatility cap of α=2 should have noimpact on margins.

Configurable Holding Period

Two notable parameters of VaR models include the length of time overwhich market risk is measured and the confidence level. The time horizonanalyzed, or the length of time determined to be required to hold theassets in the portfolio, may be referred to as the holding period. Thisholding period may be a discretionary value.

The holding period for portfolios in a risk model according to thisdisclosure may be set to be one (1) day as a default, which means onlythe risk charge to cover the potential loss for the next day isconsidered. However, due to various potential regulatory requirementsand potential changes in internal risk appetite, this value may beconfigurable to any desired value within the risk architecture describedherein. This allows for additional scenarios to be vetted under varyingrule sets. The configurable holding period can enhance the ability ofthe present disclosure to capture the risk for a longer time horizon.The following items illustrate a high level overview of thefunctionality involved:

-   -   a. the holding period, n-days, may be configured in a parameter        sheet;    -   b. the holding period value may impact returns calculations;    -   c. n-day returns, historical returns over the holding period        [e.g., ln(Price(m)/Price(m-n))] may be computed;    -   d. analytics may be performed on the n-return series;    -   e. historical price simulations may be performed over the n-day        holding period; and    -   f. profit and loss determinations may be representative of the        profit and loss over the holding period.

With a configurable holding period, the time horizon of returncalculations for both future and implied volatility (e.g., for options)may not simply be a single day. Instead, returns may be calculatedaccording to the holding period specified.

In a VaR calculation, sample overlapping is also allowed. For example,considering a three-day holding period, both the return from day one today four and the return from day two to day five may be considered to bevalid samples for the VaR calculation.

In backtesting, daily backtests may also be performed. This meansperforming backtesting for every historical day that is available forthe risk charge calculation. However, since the risk charge calculatedfor each backtesting day may have a multiple-day holding period, riskcharge may be compared to the realized profit/loss over the same timehorizon.

Notably, VaR models assume that a portfolio's composition does notchange over the holding period. This assumption argues for the use ofshort holding periods because the composition of active tradingportfolios is apt to change frequently. However, there are cases where alonger holding period is preferred, especially because it may bespecified by regulation. Additionally, the holding period can be drivenby the market structure (e.g., the time required to unwind a position inan over-the-counter (OTC) swaps market may be longer than the exchangetraded futures markets). The holding period should reflect the amounttime that is expected to unwind the risk position. Therefore, thepresent disclosure provides a model with a configurable holding period.This will allow risk management to change the holding period parameterif needed.

Expiration Model

Systems and methods of the present disclosure may be configured toprocess financial products having fixed expiries and/or relativeexpiries. Under the fixed expiry model, for each historical day,settlement data which share the same contract expiration date may beobtained to form a time series, and then historical simulation may beperformed on that series. Since the contract expiration date is fixed,as time progresses the contract will move closer to its expiry date(e.g., time to maturity is decaying). An example of a fixed time series700 is shown in FIG. 7.

On the other hand, under the relative expiry model, for each historicalday, settlement data which share the same time to maturity may be usedto form the time series. Therefore, as time progresses the contract willremain at the same distance to its expiry date and every point in thetime series may correspond to different expiration dates.

Turning now to FIG. 8, a chart 800 shows the differences betweenrelative and fixed expiry data. Any data pulled from the fixed expirymodel would fall on a fixed expiry curve. This is a curve connecting theprice points of the contract that expires at a specific time (e.g.,6/15/2012) on the forward curve day-over-day. Relative expiry data isrepresented by the curve that connects points of the contract thatexpire at a specific time period later (e.g., in one year). This figureshows that relative expiry data represents prices from severalcontracts.

Take, for example, futures contract A that issued on January '12 with aone year time to expiry. In the case where the contract is fixed expiry,data is obtained such that the contract will move closer to itsexpiration date. This implies that price changes can be tracked forcontract A by simply using the obtained data as a historical priceseries.

On the other hand, if contract A is a relative expiry contract, data isobtained such that the contract quoted will actually remain at the samedistance to its expiration date. This implies that the data consists ofquotes of different contracts with the same distance to maturity fromthe given settlement date.

Another aspect of the present disclosure is the ability to moreeffectively eliminate the seasonality impact from market risk of a givencontract. This aspect may be illustrated in the context of FIG. 9, whichshows an example chart 900 of a fixed expiry dataset. As shown in thechart 900, on every historical date (e.g. 10/15/09, 04/15/10, etc.) asystem according to this disclosure may scale the price of the samecontract (June '12). Notably, a single contract may be quoted for everyhistorical date and rather than price fluctuations caused byseasonality, these price changes may occur since the contract convergesto spot price as expiration approaches. Any significant price movementwhich deviates from the contract's natural convergence to spot, in thefixed expiry data, may be attributed to unpredictable demand or changesin economic climate.

In the context of a relative expiry dataset, the volatility in arelative expiry time series may be associated to tenor seasonalityrather than the volatility of a single tenor. As such, on everyhistorical date in such a time series, a system according to thisdisclosure may be configured to scale the price of the contract whichexpires in a constant time period away from the historical date.

Inputs into an expiration model (e.g., fixed or relative) may include,for example, a portfolio profile, historical prices and historicalvolatilities at various delta points. Under the fixed expiry model, forexample, time series data may be assumed to reference the same contract;this means that for every historical date, the data corresponding to onefixed expiration date may be obtained and will form the time series forlater scaling purposes.

An exemplary risk calculation process for calculating a risk of aproduct may include performing returns calculations which may be used inan EWMA sub-process, the results of which may be used to performstandardized return calculations and volatility forecast and capcalculations. Next, an EWMA adaptation process may be performed before aVaR calculation is performed. An option pricing process (e.g., foroptions) may also be performed following EWMA adaptation before the VaRcalculation occurs.

Option Pricing Library

Since underlying prices and option implied volatilities may be scaledseparately in an option risk charge calculation process of thisdisclosure, an option pricing library may be utilized to calculate theoption prices from scaled underlying prices and implied volatilities.Furthermore, the sticky delta technique described herein may utilizeconversions between option strike and delta, which may also achievedwithin the option pricing library.

When calculating risk charges for options, a Value-at-Risk (VaR)analysis may utilize a series of projected options values, which may notbe directly available and may therefore be calculated from scaledunderlying prices and volatilities. Therefore, the options pricinglibrary may be configured to provide an interface for the riskcalculator to price options from the scaled values and other parameters.

The sticky delta method utilizes a conversion from deltas to strikesgiven option implied volatilities for each delta after which aninterpolation can be performed in strike domain to get the interpolatedimplied volatility for further calculation.

Examples of inputs for the option pricing library may include (withoutlimitation): for ‘vanilla’ options, option type (call/put), underlyingprice, strike, time to expiry, interest rate, and implied volatility.For options on spreads, inputs may include option type (call/put),strike, first leg's underlying price, second leg's underlying price,first leg's underlying volatility, second leg's underlying volatility,time to expiry, interest rate, and implied correlation. For Asianoptions, inputs may include option type (call/put), underlying price,strike, time to expiry, interest rate, time to first averaging point,time between averaging points, number of averaging points, and impliedvolatility. For foreign exchange options, inputs may include option type(call/put), underlying price, strike, time to expiry, interest rate,interest rate of foreign currency, and implied volatility. Fordelta-to-strike conversion, input may include delta, underlying price,strike, time to expiry, interest rate, and implied volatility.

Outputs of the option pricing library may include an option price (fromoption pricer) and/or an option strike (from delta-to-strikeconversion).

Optionally, the option pricing library may be an independent moduleoutside of the risk calculation module (e.g., a standalone libraryregardless of the risk model change). In the risk calculation module, anexample function call may be as follows (function format is forillustrative purposes and could differ depending on implementation):

optionPrice=Black76Pricer(optionType, underlyingPrice, strike,timeToExpiry, interestRate, impliedVol, marginChoice) or  (13)

optionStrike=DeltaStrikeConverter(delta, underlyingPrice, strike,timeToExpiry, interestRate, impliedVol, model),  (14)

where Black-Scholes, Black76 (both margined and non-margined), spreadoption, Garmin-Kohlhagen, Barone-Adesi and Whaley, Bachelier, and Curranmodels may be implemented. Both margined and non-margined Black76 modelsmay be implemented for delta-to-strike conversion.

Yield Curve Generator (YCG)

A yield curve describes interest rates (cost of borrowing) plottedagainst time to maturity (term of borrowing). Yield curves may beutilized for pricing options because options need to be discountedcorrectly using the interest rate corresponding to their expirationdate. Also, interest rates in different countries have a directrelationship to their foreign exchange (“FX”) rates and can be used toprice forward contracts.

Yield curves may be generated daily for the settlement process by apython-based yield curve generator, which uses a data feed of OvernightIndex Swap (OIS) rates as inputs. A more generic and robust solution mayutilize similar algorithms but may be configured as a standalone moduleproviding yield curves based on client-server architecture to variousproducts.

For calculation of option value (and in turn margin), a yield curve orinterest rate corresponding to the expiry of a particular option on theday of calculation (e.g., for the Black-76 model) may be utilized.Historical interest rates to identify volatility against the strikebeing priced are also utilized. This may be accomplished by firstconverting the volatility surface from delta to strike space and theninterpolating over it. The conversion from delta space to strike spacemay utilize interest rates for the Black-76 model.

Notably, use of an interest rate may be dependent on the pricing modelbeing utilized. Thus, in a production level system which may havevarious pricing models for different instruments, interest rates may ormay not be required depending on the instrument and model used to pricethat instrument.

Inputs into a yield curve generator module may include (for example) apricing day's yield curve which for a single options contract may be theinterest rate corresponding to the expiration date of the optioncontract on the pricing date; and/or historical yield curves per VaRcalculation day, which for one options contract means the interest ratecorresponding to expiration date as of VaR calculation day which may beused for a conversion from delta to strike.

Accurate margins due to correct interest rates being used for pricingand accurate conversion from delta to strike yield proper YCG rates.

In operation, assuming the ability to query the historical interest ratecurve (yield curve) and current yield curve with granularity of time tomaturity in terms of “days” from an available database is possible, thefollowing steps can proceed:

a. Going from delta space to strike space the following formula may beused to convert delta points (e.g., seven delta points) each day intotheir corresponding strikes. Although the seven delta points in thisexample (e.g., 0.25, 0.325, 0.4, 0.5, 0.6, 0.675, and 0.75) may beconstant, the strikes corresponding to these deltas may change asunderlying shifts each day. For the Black-76 model, an equation to gofrom delta to strike space may comprise the following:

$\begin{matrix}{{K = {F*e^{- {({{{N^{- 1}{({e^{rT}*\Delta})}}*\sigma\sqrt{T}} - {\frac{\sigma^{2}}{2}*T}})}}}},} & (15)\end{matrix}$

where K=strike, F=futures price, N⁻¹=cumulative normal inverse, T=timeto expiry of option, r=interest rate, corresponding to maturity at T,σ=implied volatility and Δ=delta (change in option price per unit changein futures price).

Notably, the interest rate may be different each day in the VaR periodwhen converting from delta to strike (maturity taken with respect tocurrent day).

b. When attempting to re-price options (scenarios) using scaled datathrough an option pricing formula after EWMA volatility and underlyingprice scaling, the historical day's interest rate with maturity takenwith respect to the margining day {e.g., Today(risk calculationday)+Holding Period in business days} may be taken. The purpose here isto incorporate for interest rate risk.

c. For the backtesting process, the ability to pull historical yieldcurves to re-price options on the backtesting day may be utilized.

d. a YCG database may provide a daily yield curve with each yield curvegiving interest rates against each maturity date starting from next daywith increments of one, up to any number of years (e.g., seventy years).

Examples of option models that may be used in connection with YCGinclude (without limitation) Black-76, Margined Black-76, Spread-Li CSO,APO (Black-76) and others. Most of these models (except MarginedBlack-76) require interest rate.

Portfolio Bucketing

An aspect of the present disclosure is to calculate an initial marginfor each clearing member according to the portfolios each member holdsin their accounts. Another way to look at it is to attribute the overallmarket risk for a clearinghouse to each clearing member. However, thiswill only provide the clearinghouse's risk exposure at clearing memberlevel. Portfolio bucketing provides means for grouping clearing member'sportfolios (or divide clearing member's account) such that the riskexposure of the clearinghouse can be evaluated at a more detailed level.

There are multiple layers under each clearing member account. Thesystems and method of this disclosure may be configured to attribute aninitial margin for a clearing member to each bucketing component on eachlayer. FIG. 10 shows an exemplary clearinghouse account hierarchy 1000.As shown, the account hierarchy includes twelve (12) contract positionattributes that may pertain to a contract position within a clearingmember account. Notably, more, fewer and/or alternative attributes maybe utilized in connection with this disclosure.

For each contract position within a clearing member account, a uniquehierarchy path in the structure of FIG. 10 may be identified in order toaggregate initial margins and evaluate risk exposure at each level.After the clearing account hierarchy 1000 is built, a risk exposure toeach component (“portfolio bucket”) under the hierarchy 1000 may becalculated, with or without accounting for diversification benefitwithin the bucket. One purpose of attributing initial margin todifferent portfolio buckets is to evaluate the potential impact on theclearinghouse in scenarios where abnormal market movement occurs forcertain markets and to report initial margin for different levels ofbuckets.

As noted above, contract positions may exhibit the attributes includedin the hierarchy 1000, which may then be fed into an IRM systemaccording to this disclosure as inputs. In the exemplary hierarchy, fora particular clearinghouse 1001, the attributes include:

1. Clearing member identifier 1002 (e.g., GS, MS, JPM, etc.);

2. Trading member identifier 1003 (e.g., TM1, TM2, etc.);

3. Settlement account identifier 1004 (e.g., H, C, F, etc.);

4. Position account identifier 1005 (e.g., D, H, U, etc.);

5. Omnibus account identifier 1006 (e.g., omni1, omni2, NULL, etc.);

6. Customer account identifier 1007 (e.g., cus1, cus2, NULL, etc.);

7. Asset identifier 1008 (e.g., OIL, GAS, etc.);

8. Contingency group identifier 1009 (e.g., BrentGroup, PHEGroup, etc.);

9. Pricing group identifier 1010 (e.g., FUT, OOF, etc.)l

10. Symbol group identifier 1011 (e.g., B, BUL, H, etc.);

11. Expiration group identifier 1012 (e.g., F13, G14, Z14, etc.); and

12. Position identifier 1013.

Portfolio bucketing and initial margin aggregation will be illustratedin the context of another exemplary hierarchy 1100 shown in FIG. 11,which may apply to customer for Futures/Options (F) and customer SegFutures (W) on the settlement account level, when the customer accountsare disclosed. It may also apply to the US Customer (C) case. Thecontract position attributes for a particular clearinghouse 1101, aswell as the exemplary (non-limiting) initial margin (IM) calculations,are described below.

1. Clearing Member Level 1102:

Bucketing criteria: All the contracts that share the same clearingmember identifier may be considered to be within one clearing memberbucket.

Initial margin calculation: The initial margin attributed to eachclearing member bucket may be equal to the summation of the initialmargins attributed to all trading member account buckets under it, asshown in the equation (16) below. There may be no diversificationbenefit applied across trading member accounts.

$\begin{matrix}{{{IM}\left( {{Clearing}\mspace{14mu}{Member}\mspace{14mu} i} \right)} = {\sum\limits_{j = 0}^{n}{{IM}\left( {{{Trading}\mspace{14mu}{Member}\mspace{14mu}{Account}\mspace{14mu} j}❘{{Clearing}\mspace{14mu}{Member}\mspace{14mu} i}} \right)}}} & (16)\end{matrix}$

2. Trading Member Level 1103:

Bucketing criteria: All the contracts that share the same clearingmember identifier and trading member identifier may be considered to bewithin one trading member bucket.

Initial margin calculation: The initial margin attributed to eachtrading member bucket may be equal to the summation of the initialmargins attributed to all settlement account buckets under it, as shownin the equation (17) below. There may be no diversification benefitapplied across settlement accounts.

$\begin{matrix}{{{IM}\left( {{Trading}\mspace{14mu}{Member}\mspace{11mu} i} \right)} = {\sum\limits_{j = 0}^{n}{{IM}\left( {{{Settlement}\mspace{14mu}{Account}\mspace{14mu} j}❘{{Trading}\mspace{14mu}{Member}\mspace{14mu} i}} \right)}}} & (17)\end{matrix}$

3. Settlement Account Level 1104:

Bucketing criteria: All the contracts that share the same clearingmember identifier, trading member identifier and settlement accountidentifier may be considered to be within one settlement account bucket.

Initial margin calculation: The initial margin attributed to eachsettlement account bucket may be equal to the summation of the initialmargins attributed to all position account buckets under it, as shown inthe equation (18) below. There may be no diversification benefit appliedacross position accounts.

$\begin{matrix}{{{IM}\left( {{Settlement}\mspace{14mu}{Account}\mspace{14mu} i} \right)} = {\sum\limits_{j = 0}^{n}{{IM}\left( {{{Position}\mspace{14mu}{Account}\mspace{14mu} j}❘{{Settlement}\mspace{14mu}{Account}\mspace{14mu} i}} \right)}}} & (18)\end{matrix}$

4. Position Account Level 1105:

Bucketing criteria: All the contracts that share the same clearingmember identifier, trading member identifier, settlement accountidentifier and position account identifier may be considered to bewithin one position account bucket.

Initial margin calculation: The initial margin attributed to eachposition account bucket may be equal to the summation of the initialmargins attributed to all omnibus account buckets and all customeraccount buckets (when the customer account buckets do not belong to anyomnibus account bucket) under it, as shown in the equation (19) below.There may be no diversification benefit applied across omnibus/customeraccounts.

$\begin{matrix}{{{IM}\left( {{Position}\mspace{14mu}{Account}\mspace{14mu} i} \right)} = {\sum\limits_{j = 0}^{n}{{IM}\left( {{{{Omnibus}/{Customer}}\mspace{14mu}{Account}\mspace{14mu} j}❘{{Position}\mspace{14mu}{Account}\mspace{14mu} i}} \right)}}} & (19)\end{matrix}$

5. Omnibus Account Level 1106:

Bucketing criteria: All the contracts that share the same clearingmember identifier, trading member identifier, settlement accountidentifier, position account identifier and omnibus account identifiermay be considered to be within one omnibus account bucket.

Initial margin calculation: The initial margin attributed to eachomnibus account bucket may be equal to the summation of the initialmargins attributed to all customer account buckets under it, as shown inequation (20) below. There may be no diversification benefit appliedacross customer accounts.

$\begin{matrix}{{{IM}\left( {{Omnibus}\mspace{14mu}{Account}\mspace{14mu} i} \right)} = {\sum\limits_{j = 0}^{n}{{IM}\left( {{{Customer}\mspace{14mu}{Account}\mspace{11mu} j}❘{{Omnibus}\mspace{14mu}{Account}\mspace{11mu} i}} \right)}}} & (20)\end{matrix}$

6. Customer Account Level 1107:

Bucketing criteria: All the contracts that share the same clearingmember identifier, trading member identifier, settlement accountidentifier, position account identifier, omnibus account identifier andcustomer account identifier may be considered to be within one customeraccount bucket.

Initial margin calculation: The initial margin attributed to eachcustomer account bucket may be calculated directly from summation of theinitial margins attributed to all asset group buckets under it, as wellas the initial margin calculated from realized portfolio profit/loss(P/L), using diversification benefit calculation algorithms. Thefollowing equation (21) may apply:

$\begin{matrix}{{{IM}\left( {{Customer}\mspace{14mu}{Account}\mspace{14mu} i} \right)} = {{\quad\quad}{f_{DB}\left( {{\sum\limits_{j = 0}^{n}{{IM}\left( {{Asset}\mspace{14mu}{Group}\mspace{11mu} j} \middle| {{Customer}\mspace{14mu}{Account}\mspace{11mu} i} \right)}},{{IM}\left( {{Portfolio}\mspace{14mu} i} \right)}} \right)}}} & (21)\end{matrix}$

7. Asset Group Level 1108:

Bucketing criteria: All the contracts that share the same clearingmember identifier, trading member identifier, settlement accountidentifier, position account identifier, omnibus account identifier,customer account identifier and asset group identifier may be consideredto be within one asset group bucket.

Initial margin calculation: The initial margin attributed to each assetgroup bucket may be calculated directly from summation of the initialmargins attributed to all contingency group buckets under it, as well asthe initial margin calculated from realized portfolio P/L, usingdiversification benefit calculation algorithms. The following equation(22) may be used:

$\begin{matrix}{{{{{IM}\left( {{Asset}\mspace{14mu}{Group}\mspace{14mu} i} \right)} = {f_{DB}{\sum\limits_{j = 0}^{n}{{IM}\left( {{{Contingency}\mspace{11mu}{Group}\mspace{11mu} j}❘{{Asset}\mspace{14mu}{Group}\mspace{14mu} i}} \right)}}}},{{IM}\left( {{Portfolio}\mspace{14mu} i} \right)}}\mspace{11mu}} & (22)\end{matrix}$

8. Contingency Group Level 1109:

Bucketing criteria: All the contracts that share the same clearingmember identifier, trading member identifier, settlement accountidentifier, position account identifier, omnibus account identifier,customer account identifier, asset group identifier and contingencygroup identifier may be considered to be within one contingency groupbucket.

Initial margin calculation: The initial margin attributed to eachcontingency group may be calculated directly from the history of allpositions it contains (instead of pricing group buckets which are justone level below contingency group), allowing taking full advantage ofdiversification benefits, using the following equation (23):

IM(Contingency Group i)=f _(DB_full)(Position 1,Position 2, . . .|Contingency Group i)  (23)

9. Pricing Group Level 1110:

Bucketing criteria: All the contracts that share same clearing memberidentifier, trading member identifier, settlement account identifier,position account identifier, omnibus account identifier, customeraccount identifier, asset group identifier, contingency group identifierand pricing group identifier may be considered to be within one pricinggroup bucket.

Initial margin calculation: The initial margin attributed to eachpricing group may be calculated directly from the history of allpositions it contains (instead of symbol group buckets which are justone level below pricing group), allowing taking full advantage ofdiversification benefits, using the following equation (24):

$\begin{matrix}{{I\;{M\left( {{Pricing}\mspace{14mu}{Group}\mspace{14mu} i} \right)}} = {f_{{DB}\_{full}}\left( {{{Position}\mspace{14mu} 1},{{Position}\mspace{14mu} 2},{{.\;.\;.}\; ❘{{Pricing}\mspace{14mu}{Group}\mspace{14mu} i}}} \right)}} & (24)\end{matrix}$

10. Symbol Group Level 1111:

Bucketing criteria: All the contracts that share the same clearingmember identifier, trading member identifier, settlement accountidentifier, position account identifier, omnibus account identifier,customer account identifier, asset group identifier, contingency groupidentifier, pricing group identifier and symbol group identifier may beconsidered to be within one symbol group bucket.

Initial margin calculation: The initial margin attributed to each symbolgroup may be calculated directly from the history of all positions itcontains (instead of expiration group buckets which are just one levelbelow symbol group), allowing taking full advantage of diversificationbenefits, using the following equation (25):

$\begin{matrix}{{{IM}\left( {{Symbol}\mspace{14mu}{Group}\mspace{14mu} i} \right)} = {f_{{DB}\_{ful}l}\left( {{{Position}\mspace{14mu} 1},{{Position}\mspace{14mu} 2},{\ldots\mspace{14mu} ❘{{Symbol}\mspace{14mu}{Group}\mspace{14mu} i}}} \right)}} & (25)\end{matrix}$

11. Expiration Group Level 1112:

Bucketing criteria: All the contracts that share the same clearingmember identifier, trading member identifier, settlement accountidentifier, position account identifier, omnibus account identifier,customer account identifier, asset group identifier, contingency groupidentifier, pricing group identifier, symbol group identifier andexpiration group identifier may be considered to be within oneexpiration group bucket.

Initial margin calculation: The initial margin attributed to eachexpiration group may be calculated from the history of all positions itcontains, allowing taking full advantage of diversification benefits, asshown in by following equation (26):

$\begin{matrix}{{{IM}\left( {{Expiration}\mspace{14mu}{Group}\mspace{14mu} i} \right)} = {f_{{DB}\_{ful}l}\left( {{{Position}\mspace{14mu} 1},{{Position}\mspace{14mu} 2},{\ldots\mspace{11mu} ❘{{Expiration}\mspace{14mu}{Group}\mspace{14mu} i}}} \right)}} & (26)\end{matrix}$

12. Position Level 1113:

Bucketing criteria: Each bucket on this level only contains one singlecontract position.

Initial margin calculation: The initial margin attributed to eachposition group may be calculated as if it were a single asset portfolio.This step may form the basis of initial margin aggregation for aportfolio.

An exemplary account hierarchy 1200 of non-disclosed customer accountsis shown in FIG. 12. The hierarchy 1200 may apply to customer forFuture/Options (F) and Customer Seg Futures (W) on the settlementaccount level, when the customer accounts are non-disclosed. Thecontract position attributes for a particular clearinghouse 1201, aswell as the exemplary (non-limiting) initial margin (IM) calculations,are described below.

1. Clearing Member Level 1202:

Bucketing criteria: All the contracts that share the same clearingmember identifier may be considered to be within one clearing memberbucket.

Initial margin calculation: The initial margin attributed to eachclearing member bucket may be equal to the summation of the initialmargins attributed to all trading member account buckets under it, asshown in the following equation (27). There may be no diversificationbenefit applied across trading member accounts.

$\begin{matrix}{{{IM}\left( {{Clearing}\mspace{14mu}{Member}\mspace{14mu} i} \right)} = {\sum\limits_{j = 0}^{n}{{IM}\left( {{Trading}\mspace{14mu}{Member}\mspace{14mu}{Account}\mspace{11mu} j} \middle| {{Clearing}\mspace{14mu}{Member}\mspace{14mu} i} \right)}}} & (27)\end{matrix}$

2. Trading Member Level 1203:

Bucketing criteria: All the contracts that share the same clearingmember identifier and trading member identifier may be considered to bewithin one trading member bucket.

Initial margin calculation: The initial margin attributed to eachtrading member bucket may be equal to the summation of the initialmargins attributed to all settlement account buckets under it, as in thefollowing equation (28). There may be no diversification benefit appliedacross settlement accounts.

$\begin{matrix}{{{IM}\left( {{Trading}\mspace{14mu}{Member}\mspace{14mu} i} \right)} = {\sum\limits_{j = 0}^{n}{{IM}\left( {{Settlement}\mspace{14mu}{Account}\mspace{14mu} j} \middle| {{Trading}\mspace{14mu}{Member}\mspace{14mu} i} \right)}}} & (28)\end{matrix}$

3. Settlement Account Level 1204:

Bucketing criteria: All the contracts that share the same clearingmember identifier, trading member identifier and settlement accountidentifier may be considered to be within one settlement account bucket.

Initial margin calculation: The initial margin attributed to eachsettlement account bucket may be equal to the summation of the initialmargins attributed to all position account buckets under it, as in thefollowing equation (29). There may be no diversification benefit appliedacross position accounts.

$\begin{matrix}{{(30){{IM}\left( {{Settlement}\mspace{14mu}{Account}\mspace{14mu} i} \right)}} = {\sum\limits_{j = 0}^{n}{{IM}\left( {{Position}\mspace{14mu}{Account}\mspace{14mu} j} \middle| {{Settlement}\mspace{14mu}{Account}\mspace{14mu} i} \right)}}} & (29)\end{matrix}$

4. Position Account Level 1205:

Bucketing criteria: All the contracts that share the same clearingmember identifier, trading member identifier, settlement accountidentifier and position account identifier may be considered to bewithin one position account bucket.

Initial margin calculation: The initial margin attributed to eachposition account bucket may be equal to the summation of the initialmargins attributed to all omnibus account buckets and all customeraccount buckets (when the customer account buckets don't belong to anyomnibus account bucket) under it, as in the following equation (30).There may be no diversification benefit applied across omnibus/customeraccounts.

$\begin{matrix}{{{IM}\left( {{Position}\mspace{14mu}{Account}\mspace{14mu} i} \right)} = {\sum\limits_{j = 0}^{n}{{IM}\left( {{{Omnibus}/{Customer}}\mspace{14mu}{Account}\mspace{14mu} j} \middle| {{Position}\mspace{14mu}{Account}\mspace{14mu} i} \right)}}} & (30)\end{matrix}$

5. Omnibus Account Level 1206:

Bucketing criteria: All the contracts that share the same clearingmember identifier, trading member identifier, settlement accountidentifier, position account identifier and omnibus account identifiermay be considered to be within one omnibus account bucket.

Initial margin calculation: The initial margin attributed to eachomnibus account bucket may be equal to the initial margins of thenon-disclosed customer account buckets under it, as in the followingequation (31).

$\begin{matrix}{{{IM}\left( {{Omnibus}\mspace{14mu}{Account}\mspace{14mu} i} \right)} = {{IM}\left( {{Non} - {{disclosed}\mspace{14mu}{Customer}\mspace{14mu}{Account}\mspace{14mu} i}} \right)}} & (31)\end{matrix}$

6. Customer Account Level 1207:

Bucketing criteria: All the contracts that share the same clearingmember identifier, trading member identifier, settlement accountidentifier, position account identifier, omnibus account identifier andcustomer account identifier may be considered to be within one customeraccount bucket.

Initial margin calculation: The initial margin attributed to eachcustomer account (non-disclosed) bucket may be equal to the summation ofthe initial margins attributed to all asset group buckets under it, asin the following equation (32). There may be no diversification benefitapplied across omnibus/customer accounts.

$\begin{matrix}{{{{IM}\left( {{Customer}\mspace{14mu}{Account}\mspace{14mu} i} \right)} = {\sum\limits_{j = 0}^{n}{{IM}\left( {{{Asset}\mspace{14mu}{Group}\mspace{14mu} j}❘{{Customer}\mspace{14mu}{Account}\mspace{14mu} i}} \right)}}}\;} & (32)\end{matrix}$

7. Asset Group Level 1208:

Bucketing criteria: All the contracts that share the same clearingmember identifier, trading member identifier, settlement accountidentifier, position account identifier, omnibus account identifier,customer account identifier and asset group identifier may be consideredto be within one asset group bucket.

Initial margin calculation: The initial margin attributed to each assetgroup bucket may be equal to the summation of the initial marginsattributed to all contingency group buckets under it, as in thefollowing equation (33). There may be no diversification benefit appliedacross contingency group buckets.

$\begin{matrix}{{{IM}\left( {{Asset}\mspace{14mu}{Group}\mspace{14mu} i} \right)} = {\sum\limits_{j = 0}^{n}{{IM}\left( {{{Contingency}\mspace{14mu}{Group}\mspace{14mu} j}❘{{Asset}\mspace{14mu}{Group}\mspace{14mu} i}} \right)}}} & (33)\end{matrix}$

8. Contingency Group Level 1209:

Bucketing criteria: All the contracts that share the same clearingmember identifier, trading member identifier, settlement accountidentifier, position account identifier, omnibus account identifier,customer account identifier, asset group identifier and contingencygroup identifier may be considered to be within one contingency groupbucket.

Initial margin calculation: The initial margin attributed to eachcontingency group may be equal to the summation of the initial marginsattributed to all pricing group buckets under it, as in the followingequation (34). There may be no diversification benefit applied acrosspricing group buckets.

$\begin{matrix}{{{IM}\left( {{Contingency}\mspace{14mu}{Group}\mspace{14mu} i} \right)} = {\sum\limits_{j = 0}^{n}{{IM}\;\left( {{{Pricing}\mspace{14mu}{Group}\mspace{14mu} j}❘{{Contingency}\mspace{14mu}{Group}\mspace{14mu} i}} \right)}}} & (34)\end{matrix}$

9. Pricing Group Level 1210:

Bucketing criteria: All the contracts that share same clearing memberidentifier, trading member identifier, settlement account identifier,position account identifier, omnibus account identifier, customeraccount identifier, asset group identifier, contingency group identifierand pricing group identifier may be considered to be within one pricinggroup bucket.

Initial margin calculation: The initial margin attributed to eachpricing group may be equal to the summation of the initial marginsattributed to all symbol group buckets under it, as in the followingequation (35). There may be no diversification benefit applied acrosssymbol group buckets.

$\begin{matrix}{{{IM}\left( {{Pricing}\mspace{14mu}{Group}\mspace{14mu} i} \right)} = {\sum\limits_{j = 0}^{n}{{IM}\left( {{{Symbol}\mspace{14mu}{Group}\mspace{14mu} j}❘{{Pricing}\mspace{14mu}{Group}\mspace{14mu} i}} \right)}}} & (35)\end{matrix}$

10. Symbol Group Level 1211:

Bucketing criteria: All the contracts that share the same clearingmember identifier, trading member identifier, settlement accountidentifier, position account identifier, omnibus account identifier,customer account identifier, asset group identifier, contingency groupidentifier, pricing group identifier and symbol group identifier may beconsidered to be within one symbol group bucket.

Initial margin calculation: The initial margin attributed to each symbolgroup may be equal to the summation of the initial margins attributed toall expiration group buckets under it, as in the following equation(36). There may be no diversification benefit applied across expirationgroup buckets.

$\begin{matrix}{{{IM}\left( {{Symbol}\mspace{14mu}{Group}\mspace{14mu} i} \right)} = {\sum\limits_{j = 0}^{n}{{IM}\left( {{{Expiration}\mspace{14mu}{Group}\mspace{14mu} j}❘{{Symbol}\mspace{14mu}{Group}{\;\mspace{11mu}}i}} \right)}}} & (36)\end{matrix}$

11. Expiration Group Level 1212:

Bucketing criteria: All the contracts that share the same clearingmember identifier, trading member identifier, settlement accountidentifier, position account identifier, omnibus account identifier,customer account identifier, asset group identifier, contingency groupidentifier, pricing group identifier, symbol group identifier andexpiration group identifier may be considered to be within oneexpiration group bucket.

Initial margin calculation: The initial margin attributed to eachexpiration group may be equal to the summation of the initial marginsattributed to all position group buckets under it, as in the followingequation (37). There may be no diversification benefit applied acrossposition group buckets.

$\begin{matrix}{{{IM}\left( {{Expiration}\mspace{14mu}{Group}\mspace{14mu} i} \right)} = {\sum\limits_{j = 0}^{n}{{IM}\left( {{{Position}\mspace{14mu}{Group}\mspace{14mu} j}❘{{Expiration}\mspace{14mu}{Group}\mspace{14mu} i}} \right)}}} & (37)\end{matrix}$

12. Position Level 1213:

Bucketing criteria: Each bucket on this level may only contain onesingle contract position.

Initial margin calculation: The initial margin attributed to eachposition group may be calculated as if it were a single asset portfolio.

Additional exemplary account hierarchies 1300, 1400 are shown in FIGS.13 and 14, respectively. These hierarchies may apply to house accounts(H) and non-US customers (C) on the settlement account level. Thecontract position attributes for clearinghouses 1301, 1401 as well asthe exemplary (non-limiting) initial margin (IM) calculations, aredescribed below.

1. Clearing Member Level 1302, 1402:

Bucketing criteria: All the contracts that share the same clearingmember identifier may be considered to be within one clearing memberbucket.

Initial margin calculation: The initial margin attributed to eachclearing member bucket may be equal to the summation of the initialmargins attributed to all trading member account buckets under it, as inthe following equation (38). There may be no diversification benefitapplied across trading member accounts.

$\begin{matrix}{{{IM}\left( {{Clearing}\mspace{14mu}{Member}\mspace{14mu} i} \right)} = {\sum\limits_{j = 0}^{n}{{IM}\left( {{{Trading}\mspace{14mu}{Member}\mspace{14mu}{Account}\mspace{14mu} j}❘{{Clearing}\mspace{14mu}{Member}\mspace{14mu} i}} \right)}}} & (38)\end{matrix}$

2. Trading Member Level 1303, 1403:

Bucketing criteria: All the contracts that share the same clearingmember identifier and trading member identifier may be considered to bewithin one trading member bucket.

Initial margin calculation: The initial margin attributed to eachtrading member bucket may be equal to the summation of the initialmargins attributed to all settlement account buckets under it, as in thefollowing equation (39). There may be no diversification benefit appliedacross settlement accounts.

$\begin{matrix}{{{IM}\left( {{Trading}\mspace{14mu}{Member}\mspace{14mu} i} \right)} = {\sum\limits_{j = 0}^{n}{{IM}\left( {{{Settlement}\mspace{14mu}{Account}\mspace{14mu} j}❘{{Trading}\mspace{14mu}{Member}\mspace{14mu} i}} \right)}}} & (39)\end{matrix}$

3. Settlement Account Level 1304, 1404:

Bucketing criteria: All the contracts that share the same clearingmember identifier, trading member identifier and settlement accountidentifier may be considered to be within one settlement account bucket.

Initial margin calculation: The initial margin attributed to eachsettlement account bucket may be equal to the summation of the initialmargins attributed to all position account buckets under it, as in thefollowing equation (40). There may be no diversification benefit appliedacross position accounts.

$\begin{matrix}{{{IM}\left( {{Settlement}\mspace{14mu}{Account}\mspace{14mu} i} \right)} = {\sum\limits_{j = 0}^{n}{{IM}\left( {{{Position}\mspace{14mu}{Account}\mspace{14mu} j}❘{{Settlement}\mspace{14mu}{Account}\mspace{14mu} i}} \right)}}} & (40)\end{matrix}$

4. Position Account Level 1305, 1405:

Bucketing criteria: All the contracts that share the same clearingmember identifier, trading member identifier, settlement accountidentifier and position account identifier may be considered to bewithin one position account bucket.

Initial margin calculation: The initial margin attributed to eachposition account bucket may be calculated directly from summation of theinitial margins attributed to all asset group buckets under it, as wellas the initial margin calculated from realized portfolio P/L, usingdiversification benefit calculation algorithms. An exemplary equation(41) is below:

$\begin{matrix}{{{IM}\left( {{Position}\mspace{14mu}{Account}\mspace{14mu} i} \right)} = {f_{DB}\left( {{\sum\limits_{j = 0}^{n}{{IM}\left( {{Asset}\mspace{14mu}{Group}\mspace{14mu} j} \middle| {{Position}\mspace{14mu}{Account}\mspace{14mu} i} \right)}},{{IM}\left( {{Portfolio}\mspace{14mu} i} \right)}} \right)}} & (41)\end{matrix}$

5. Customer Account Level 1307, 1407: n/a.

6. Asset Group Level 1308, 1408:

Bucketing criteria: All the contracts that share the same clearingmember identifier, trading member identifier, settlement accountidentifier, position account identifier and asset group identifier maybe considered to be within one asset group bucket.

Initial margin calculation: The initial margin attributed to each assetgroup bucket may be calculated directly from summation of the initialmargins attributed to all contingency group buckets under it, as well asthe initial margin calculated from realized portfolio P/L, usingdiversification benefit calculation algorithms. An exemplary equation(42) is provided below.

$\begin{matrix}{{{IM}\left( {{Asset}\mspace{14mu}{Group}\mspace{14mu} i} \right)} = {f_{DB}\left( {{\sum\limits_{j = 0}^{n}{{IM}\left( {{Contingency}\mspace{14mu}{Group}\mspace{14mu} j} \middle| {{Asset}\mspace{14mu}{Group}\mspace{14mu} i} \right)}},{{IM}\left( {{Portfolio}\mspace{14mu} i} \right)}} \right)}} & (42)\end{matrix}$

7. Contingency Group Level 1309, 1409:

Bucketing criteria: All the contracts that share the same clearingmember identifier, trading member identifier, settlement accountidentifier, position account identifier, asset group identifier andcontingency group identifier may be considered to be within onecontingency group bucket.

Initial margin calculation: The initial margin attributed to eachcontingency group may be calculated directly from the history of allpositions it contains (instead of pricing group buckets which are justone level below contingency group), as in the following equation (43),allowing taking full advantage of diversification benefits.

$\begin{matrix}{{{IM}\left( {{Contingency}\mspace{14mu}{Group}\mspace{14mu} i} \right)} = {f_{{DB}\_{ful}l}\left( {{{Position}\mspace{14mu} 1},{{Position}\mspace{14mu} 2},{\ldots\mspace{14mu} ❘{{Contingency}\mspace{14mu}{Group}\mspace{14mu} i}}} \right)}} & (43)\end{matrix}$

8. Pricing Group Level 1310, 1410:

Bucketing criteria: All the contracts that share same clearing memberidentifier, trading member identifier, settlement account identifier,position account identifier, asset group identifier, contingency groupidentifier and pricing group identifier may be considered to be withinone pricing group bucket.

Initial margin calculation: The initial margin attributed to eachpricing group may be calculated directly from the history of allpositions it contains (instead of symbol group buckets which are justone level below pricing group), as in the following equation (44),allowing taking full advantage of diversification benefits.

$\begin{matrix}{{{IM}\left( {{Pricing}\mspace{14mu}{Group}\mspace{14mu} i} \right)} = {f_{{DB}\_{full}}\left( {{{Position}\mspace{14mu} 1},{{Position}\mspace{14mu} 2},{\ldots\mspace{14mu} ❘{{Pricing}\mspace{14mu}{Group}\mspace{14mu} i}}} \right)}} & (44)\end{matrix}$

9. Symbol Group Level 1311, 1411:

Bucketing criteria: All the contracts that share the same clearingmember identifier, trading member identifier, settlement accountidentifier, position account identifier, asset group identifier,contingency group identifier, pricing group identifier and symbol groupidentifier may be considered to be within one symbol group bucket.

Initial margin calculation: The initial margin attributed to each symbolgroup may be calculated directly from the history of all positions itcontains (instead of expiration group buckets which are just one levelbelow symbol group), as in the following equation (45), allowing takingfull advantage of diversification benefits.

$\begin{matrix}{{{IM}\left( {{Symbol}\mspace{14mu}{Group}\mspace{14mu} i} \right)} = {f_{{DB}_{full}}\left( {{{Position}\mspace{14mu} 1},{{Position}\mspace{14mu} 2},{\ldots\mspace{14mu} ❘{{Symbol}\mspace{14mu}{Group}\mspace{14mu} i}}} \right)}} & (45)\end{matrix}$

10. Expiration Group Level 1312, 1412:

Bucketing criteria: All the contracts that share the same clearingmember identifier, trading member identifier, settlement accountidentifier, position account identifier, asset group identifier,contingency group identifier, pricing group identifier, symbol groupidentifier and expiration group identifier may be considered to bewithin one expiration group bucket.

Initial margin calculation: The initial margin attributed to eachexpiration group may be calculated from the history of all positions itcontains, as in the following equation (46), allowing taking fulladvantage of diversification benefits.

$\begin{matrix}{{{IM}\left( {{Expiration}\mspace{14mu}{Group}\mspace{14mu} i} \right)} = {f_{{DB}\_{full}}\left( {{{Position}\mspace{14mu} 1},{{Position}\mspace{14mu} 2},{\ldots\mspace{14mu} ❘{{Expiration}\mspace{14mu}{Group}\mspace{14mu} i}}} \right)}} & (46)\end{matrix}$

11. Position Level 1313, 1413:

Bucketing criteria: Each bucket on this level may only contains onesingle contract position.

Initial margin calculation: The initial margin attributed to eachposition group may be calculated as if it were a single asset portfolio.

Diversification Benefit

A diversification benefit process according to this disclosure mayassume the following: 1) a customer's account may be considered aportfolio with a natural hierarchy 1500, as shown in FIG. 15; each levelof the hierarchy 1500 may have pairwise diversification benefitcoefficients defined at each level, which may by default be set up tohave zero haircut, meaning no affect on VaR margin; and a haircut may beapplied to the diversification benefit at each level (for any reasondeemed necessary or desirable).

In summary, an exemplary process for determining IM that accounts fordiversification benefit may include one or more of the followingexemplary steps:

-   -   a. compute a separate and combined margin at each level of the        hierarchy for a customer account;    -   b. compute a diversification benefit at each level of the        hierarchy;    -   c. perform a diversification attribution at each level of the        hierarchy;    -   d. inside each level, the diversification benefit may be        allocated to each possible pair;    -   e. the diversification benefit coefficient may then be used to        haircut the diversification benefit given to the customer        account;    -   f. the sum of the diversification benefit haircuts at each level        may then be added on to the fully diversified margin charge; and    -   g. the haircuts across each level of the hierarchy may be added        to arrive at an initial margin.

Referring again to FIG. 15, components of the exemplary hierarchy 1500may include levels such as account 1501, sector 1502, contingency group1503, product 1504 and position 1505. The account level 1501 is shown asthe topmost level, which may be the level at which a final initialmargin may be reported. An account 1501 may be made up of sectors 1502.

Sectors 1502 may be made up of contingency groups such as, for example,North American Power, North American Natural Gas, UK Natural Gas,European Emissions, etc.

The contingency group 1503 level may include collections of productsthat may have direct pricing implications on one another. For example,an option on a future and the corresponding future. An example of acontingency group (CG) may be a Brent={B, BUL, BRZ, BRM, . . . }, i.e.,everything that ultimately refers to Brent crude as an underlying forderivative contracts. Contingency groups 1503 may be composed ofproducts.

The product level 1504 may include groups of products, includingphysical or financial claims on a same (physical or financial)underlying. Non-limiting examples of products include Brent Futures,Options on WTI futures, AECO Natural Gas Basis swaps, etc.

The position level 1505 may comprise distinct positions in a clearedcontracts within a customer's account. Non-limiting examples ofpositions may be referred to as 100 lots in Brent Futures, −50 lots inOptions on WTI futures, and −2,500 lots in AECO Basis Swaps, etc.

Concepts associated with diversification benefit and the hierarchydiscussed above are provided below. Notably, some of the following termsand concepts have already been discussed above. The followingdescriptions are intended to supplement (and not limit) any of thedescriptions provided above.

Margin may be used interchangeably with initial margin, discussed above.Margin may be the amount of capital required to collateralize potentiallosses from the liquidation of a customer's portfolio over an assumedholding period and to a particular statistical confidence interval.

$\begin{matrix}{{M_{1} = {{Margin}\;\left( {{Position}\mspace{14mu} 1} \right)}},{M_{1,2} = {{Margin}\mspace{11mu}\left( {{{Position}\mspace{14mu} 1} + {{Position}\mspace{14mu} 2}} \right)}},} & (47)\end{matrix}$

where i and j refer to indices across all sectors so Mi may be themargin for the ith sectors and Mi,j refers the margin of a pairwisecombined sectors.

Margin Separate (Msep) refers to the sum of the margins calculated onevery position's individual profits and loss array in a portfolio. Thisrefers to a worst possible case in which there is no diversificationbenefit.

$\begin{matrix}{{Msep} = {{M\left( {{Position}\mspace{14mu} 1} \right)} + {M\left( {{Position}\mspace{14mu} 2} \right)} + {M\left( {{Position}\mspace{14mu} 3} \right)} + \ldots}} & (48)\end{matrix}$

Margin Combined (Mcomb) refers to a margin calculated on an entireportfolio's profit and loss array. This is the case in which fulldiversification benefit is given.

$\begin{matrix}{{Mcomb} = {M\left( {{{Position}\mspace{14mu} 1} + {{Position}\mspace{14mu} 2} + {{Position}\mspace{14mu} 3\ldots}}\; \right)}} & (49)\end{matrix}$

This may also be referred to as a fully diversified margin.

Offset may refer to a decrease in margin due to portfoliodiversification benefits.

$\begin{matrix}{{Offset} = \left( {{Msep} - {Mcomb}} \right)} & (50)\end{matrix}$

Haircut may refer to a reduction in the diversification benefit.

Diversification Benefit (DB) may refer to a theoretical reduction inrisk a portfolio achieved by increasing the breadth of exposures tomarket risks over the risk to a single exposure; based, for example,upon a Markowitz portfolio theory. In the context of this disclosure, adiversification benefit (DB) may be a metric of the risk measurereduction an account receives by viewing risk from a portfolioperspective versus a position perspective.

$\begin{matrix}{{DB} = {{Msep} - {Mcomb}}} & (51)\end{matrix}$

Rearranging this equation provides:

$\begin{matrix}{{Mcomb} = {{Msep} - {DB}}} & (52)\end{matrix}$

In this way, the DB may be defined as a “dollar” value.

Diversification Benefit Coefficient (y) may be a number between zero (0)and one (1) that indicates an amount of diversification benefit allowedfor an account. Conceptually, a diversification benefit of zero maycorrespond to the sum of the margins for sub-portfolios, while adiversification benefit of one may be the margin calculated on the fullportfolio.

$\begin{matrix}{{Mnew} = {{Msep} - {y*{DB}}}} & (53)\end{matrix}$

According to the foregoing equation, Mnew may be equal to Msep or Mcombby setting y equal to zero or one respectively.

Diversification Benefit Haircut (h) may refer to the amount of thediversification benefit charged to an account, representing a reductionin diversification benefit. If the subscripts for y_(ij) refer to thesub-portfolios, the diversification benefit haircut can be expressed asone (1) minus y_(ij), as in the following equation:

$\begin{matrix}{h_{1,2} = {1 - y_{1,2}}} & (54)\end{matrix}$

This may be the pairwise margin haircut.

Margin Offset Contribution (OC) may refer to the margin offsetcontribution of combining multiple instruments into the same portfolioversus margining them separately. The offset contribution for a pair ofproducts may be the diversification benefit for that set of portfolios:

$\begin{matrix}{{OC}_{1,2} = {M_{1} + M_{2} - M_{1,2}}} & (55) \\{{OC}_{1,3} = {M_{1} + M_{3} - M_{1,3}}} & (56) \\{{OC}_{2,3} = {M_{2} + M_{3} - M_{2,3}}} & (57)\end{matrix}$

Offset Ratio (OR) may refer to the ratio of total portfoliodiversification benefit to the sum of pairwise diversification benefits.

$\begin{matrix}{{DB_{portfolio}} = {{M_{1} + M_{2} + M_{3} - M_{1,2,3}} = {M_{sep} - M_{comb}}}} & (59) \\{{OR} = \frac{DB_{portfolio}}{\left( {{OC}_{1,2} + {OC}_{1,3} + {OC}_{2,3}} \right)}} & (60)\end{matrix}$

This ratio forces the total haircut to be no greater than the sum ofoffsets at each level.

Haircut Weight (w) may refer to the percentage of a margin offsetcontribution that will be the haircut at each level.

$\begin{matrix}{w_{1,2} = {h_{1,2}*{OR}}} & (61) \\{w_{1,3} = {h_{1,3}*{OR}}} & (62) \\{w_{2,3} = {h_{2,3}*{OR}}} & (63)\end{matrix}$

Haircut Contribution (HC) may refer to the contribution to thediversification haircut for each pair at each level.

$\begin{matrix}{{HC_{1,2}} = {w_{1,2}*{OC}_{1,2}}} & (64) \\{{HC_{1,3}} = {w_{1,3}*{OC}_{1,3}}} & (65) \\{{HC_{2,3}} = {w_{2,3}*{OC}_{2,3}}} & (66)\end{matrix}$

Level Haircut may refer to the haircut at each level.

$\begin{matrix}{{Haircut} = {{HC_{1,2}*HC_{1,3}} + {HC_{2,3}}}} & (67)\end{matrix}$

IRM Margin may refer to an actual margin charge.

$\begin{matrix}{{M\_{IRM}} = {{Mcomb} + {Haircut}}} & (68)\end{matrix}$

An exemplary diversification benefit process according to thisdisclosure may include the following process steps.

1. Computing base margins, which result in the Mcomb and Msep marginamounts at each level in the account hierarchy. Elements of computingbase margins may include identifying a financial portfolio; identifyingdiversification benefit coefficients associated with the financialportfolio; computing instrument level margins, position level marginsand rolling up the positions to compute fully diversified margins foreach product separately (i.e., product level margin computation);computing contingency level margins by combining the products into acontingency group level to compute fully diversified margins for eachproduct group separately; computing sector level margins by combiningthe contingency groups in order to compute fully diversified margins foreach sector separately; and computing an account margin by combining thesectors to compute fully diversified margins for an overall account(e.g., a customer account).

2. computing fully diversified margins across all account levels;

3. computing a margin offset across all account levels; and

4. computing diversification haircuts, by: computing inter-sectordiversification haircuts; computing inter-contingency groupdiversification haircuts; computing inter-product diversificationhaircuts; computing inter-month diversification haircuts; and computingtotal IRM margin diversification haircuts.

Information from this foregoing process may then be compiled into a“dashboard” (which may be displayed via an interactive GUI).

Next, systems and methods for more efficiently determining an initialmargin are described, according to another embodiment of the presentdisclosure. A risk management system, according to some embodiments, mayinclude a model system configured to determine an initial margin in anefficient manner while also capturing any liquidity-related risks. Therisk management system may also be configured to perform modelcalibration and model testing. The risk management system may beconfigured to perform model testing and calibration not only with actualhistorical data, but also through synthetic data. Moreover, thesynthetic data may be created to model both benign data conditions aswell as extreme events, such as one or more regime changes. In thismanner one or more models of the model system may be created and updatedto provide stable and accurate risk data for one or more financialproducts over a variety of data source conditions.

In some examples, the model system of the risk management system mayinclude a margin model and a liquidity risk charge (LRC) model. Themargin model (also referred to as IRM), in general, may be configured toestimate the risk of future losses, by generating an initial margindetermination. The LRC model, in general, may be configured to assess aliquidity risk at the product portfolio level. Notably, the LRC modelmay be configured to quantify liquidity-related risks that may not becapable of being captured in the initial margin determination (by themargin model).

In general, the margin model may be configured to determine anappropriate amount for initial margin, in order to protect theclearinghouse in case of member default. In the regular course oftransactions, where all positions are matched, the clearinghouse doesnot assume market price risk. This is because all of its long positionshave an identical short position as financial market participants arematched through the clearinghouse. While the clearinghouse may notassume market risk, the clearinghouse may assume the credit risk thatcomes with acting as the central counterparty (CCP). CCPs are configuredas risk managers. CCPs serve as a substituted counterparty to both sidesof a transaction brought to them by their clearing members. In contrastto the mismatched books of banks, CCPs run matched books. They do nottake risk other than counterparty risk in connection with their role assubstituted counterparty. They assume position risk only when a memberfails to perform its obligations until such time as the position risk iseither liquidated or transferred to another member.

Initial margin may be used in the event of default, to cover the riskthe CCP instantly takes on when it assumes the positions of itsdefaulted clearing member. The risk may be collateralized to a highdegree of certainty (e.g., 99%) in the profit and loss distribution ofthe defaulted portfolio.

However, charging additional initial margin in times of market distresscould lead to additional market distress. To avoid such an outcome, theCCP is typically careful not to create additional systemic risk, byapplying buffers in times of market calm. This then permits the CCP tocarry a risk factor buffer that can be released in times of marketdistress. Such a risk factor buffer may be achieved within the marginmodel of the present disclosure, by adhering to anti-pro-cyclical marginrequirements.

At a high level, the margin model may be configured to use historicalprice fluctuations (“volatility”) to estimate the risk of future losses.By using historical volatility, the margin model may predict that, witha high degree of certainty (e.g., at least 99%), losses due to adversemarket price movements, over a given period of time for a givenportfolio (e.g., a portfolio of interest rate derivatives), may notexceed (in magnitude) the initial margin requirements. As an example, ifthe output of the margin model is −$1 million, this means that with atleast 99% certainty from the time of a clearing member default until theend of a projected liquidation period (e.g., two days), adverse marketprice changes can cause losses of up to $1 million. Collateralization ofthe $1 million in initial margin prior to default may protect theclearinghouse system against potential losses.

Conceptually, the margin model, according to some embodiments, makes afundamental risk management proposition that risk may be desirablyquantified and managed from a portfolio perspective. For example, a moreefficient allocation of capital may be achieved by formally permittingrisk offsets where appropriate, while maintaining an overall riskprofile that offers sufficient protection for the clearinghouse system.More technically, the margin model may be configured to generatevolatility forecasts from historical returns and estimate a jointdistribution of portfolio level profit and loss (P&L, also referred toas PnL herein), in a non-parametric way and without formally estimatingthe covariance matrix. In some embodiments, the margin model may alsoinclude volatility components, such as a volatility floor, as part of adetermination of a volatility forecast.

In some embodiments, the margin model may comprise a statistical model,and may be configured in accordance with a set of assumptions (e.g.,with respect to data, methodology, model parameter selection, andtesting). In an example embodiment, assumptions of the margin model mayinclude assumptions related to portfolio margining, filtering, timescaling, procyclicality risk and calibration methodology.

With respect to portfolio margining, the margin model assumes that anyoffsets in the margin model may be based on significant and reliablecorrelations in the data. As discussed further below, the margin modelmay mitigate any risk associated with this assumption based on acorrelation stress component.

With respect to filtering, the margin model may perform filtering usingexponentially weighted moving average (EWMA) volatilities. Morespecifically, the margin model may be configured to perform an EWMAprocess to generate volatility forecasts from historical returns. Everyhistorical return may be divided by historical volatility and thenmultiplied by the forecast volatility (“scaled”). The margin modelassumes that recent volatility has higher information content forpredicting future volatility when compared to volatility experiencedless recently.

With respect to time scaling, the margin model may be configured toachieve a multiday holding period time scaling by using a“square-root-of-time” rule. For example, and as discussed further below,the margin model may first generate a set of historical one-day returns.To achieve a two-day holding period, the one day returns may bemultiplied by the square root of 2. Through, the use of the square rootof time rule, the margin model assumes the one-day returns areuncorrelated and have constant variance over the holding period.

With respect to pro-cyclicality risk, the margin model may be configuredto mitigate any pro-cyclicality risk by an anti-pro-cyclicality (APC)treatment process. The APC treatment may be configured to reducepro-cyclicality, by moderating large step changes in initial margin whenmargin requirements are rising significantly.

With respect to the calibration methodology, the calibration methods ofthe margin model may be chosen for particular model parameters, such asan EWMA weight and correlation stress weight (described further below).Accordingly the margin model is assumed to be calibrated according to amethodology that is theoretically sound and such that the calibratedvalues are robust.

In some embodiments, historical correlation may be largely maintained aspart of the scaling process of the margin model, however the marginmodel may not explicitly forecast correlation. Instead, the margin modelmay determine a risk of declining correlation (through a correlationstress component, described further below).

As discussed above, in the event of clearing member default, the CCP mayassume ownership of any instrument belonging to the respective accountsof the clearing member. Any positions assumed by the CCP may then beclosed out (liquidated), auctioned or held to expiry, depending on whatis deemed the best course of action at that time. One purpose of the LRCmodel is to ensure that the CCP collects upfront a margin amountsufficient to protect against the risk of an adverse financial impactdue to relatively large trading positions as well any bid-ask spreadcosts that may be incurred as part of closing positions following one ormore default scenarios. In this capacity, the LRC model may account forrisk not captured in the margin model.

In some embodiments, LRC values generated by the LRC model may representan additional overnight requirements for the relevant accounts. In someexamples, the LRC values may be recalculated daily, due to the dynamicnature of the portfolios as well as updated market pricing. In someexamples, each day new requirements including LRC values may be passedon to a banking system, ready for an end of margin call process mandatedfor all accounts held at the CCP.

More specifically, the margin model of the present disclosure, in someexamples, may be configured to evaluate a price risk at a settlementprice level (hence without accounting for possible bid-ask spread costs)from a beginning to an end of a holding period, or a margin period ofrisk (MPOR). The CCP may then be protected, up to a specified level ofconfidence, against adverse market price changes in the event of one ormore clearing member (CM) defaults. In some embodiments, the LRC modelmay be configured to quantify liquidity-related risks that may not becaptured in the initial margin requirement. In some examples, positionsthat are large relative to typical transactions in a given market maytake longer to unwind than is anticipated in the MPOR of the initialmargin. The LRC model may be configured to quantify an amount tocollateralise for mitigating this exposure.

The LRC model may be configured to assess the following charges to coverpossible liquidity risks: a concentration charge (CC) and a bid-askcharge (BAC). The concentration charge may represent a margin add-onvalue from clearing members of the clearinghouse system to protect theCCP from liquidation costs associated with liquidating large clearedpositions. The bid-ask charge may represent a margin add-on value fromclearing members to protect the CCP from liquidation costs associatedwith the bid-ask spread.

From a high level, the LRC model may assess liquidity risk at theportfolio level (as opposed to at the individual instrument level). Inthis manner the LRC model may extend the notion of portfolio marginingto liquidity risk (albeit at instrument group level) so that feweroffsets may be provided compared to the margin model. Intuitively, whereindividual instruments may be less liquid, to the extent price risk islow at their combined portfolio level, the CCP faces a reduced exposure.

More technically, the LRC model may be configured to construct pricerisk equivalent portfolio representations by using representativeinstruments. In some examples, whether a position in a representativeinstrument is concentrated may be determined by predeterminedconcentration thresholds calibrated based on the volume and openinterest data. If the position is expected to take longer to liquidatethan the MPOR of the initial margin, the concentration charge may beapplied based on the excess positions.

The BAC makes up the second component of the LRC model and also appliesat the instrument group level. In some examples, the BAC may not bebased on an excess quantity beyond a predetermined threshold, but rathermay be based on an entire position (e.g., in the representativeinstrument) that qualifies for BAC. In some examples, the LRC model maybe configured to output a sum of the concentration charge and thebid-ask charge, to form a (quantified) liquidity risk charge value.

As with the margin model, in some examples, the LRC model may also beconfigured according to various assumptions. For examples, the LRC modelmay assume that the use of portfolio representations provides for anadequate way of estimating the liquidity risk charge. For example, theLRC model may be configured to determine a position in a representativeinstrument such that the price risk is equivalent to the set ofinstruments represented. As another example, the concentration chargemay be determined under the assumption that positions are graduallyliquidated without impacting the market, and adequately captures theliquidity risk of a portfolio.

Similar to the model described above with respect to FIGS. 1-15, themargin model described below also utilizes a VaR as the risk measure.The margin model described below also determines the initial marginbased on historical (raw) risk factor returns; scaling of historicalrisk factor returns by volatility to generate risk factor scenariosbased on current risk factor values; and volatility forecasting based onEWMA. Moreover, the model system described below may also be configuredto efficiently and accurately determine risk data for both linearproducts (e.g., futures) and non-linear financial products (e.g.,options) by an empirical approach, by using a historical simulationprocedure.

The margin model described below is different from the above-describedmodel, in that the margin model does not apply correlation scaling onrisk factors. Instead, in order to account for any correlation change,the margin model (described below) includes a correlations stresscomponent as a buffer for any sudden increase or decrease across riskfactors. Another difference is with respect to the MPOR. The modeldescribed above may apply scaling on overlapping n-day returns (with nbeing equal to MPOR). The margin model described below does not apply ascaling on overlapping n-day returns. Instead, the margin modeldescribed below starts with one-day returns (therefore beingnon-overlapping) and scales up the volatility forecast with a squareroot factor. Moreover, unlike the model described above with respect toFIGS. 1-15, the model system described below includes the LRC model toassess liquidity risk.

In addition, the margin model of the present disclosure, describedbelow, includes additional components such as a volatility floor, astress volatility component, an APC treatment, a portfoliodiversification benefit limit and currency allocation. The volatilityfloor may provide a minimum value for the forecasted volatility of eachrisk factor, to ensure an overall conservative margin estimate in a lowmarket volatility environment. The stress volatility componentrepresents an additional component in addition to the volatilityforecast, to ensure that periods of market stress are included for thecalculating of volatilities. The APC treatment may be configured todynamically adjust the volatility forecast based on recent market moves,which generally may reduce the margin pro-cyclicality. The portfoliodiversification benefit limit may comprise a measure to limit an amountof portfolio offsets for multi-product portfolios (e.g., in accordancewith particular regulations). In some examples, the portfoliodiversification benefit limit may aid in mitigating a risk that thecorrelation to be realized in the future over the margin period turnsout to be “worse” (e.g., riskier) than what may be contained in thehistorical data. The currency allocation may be configured to allocatethe initial margin to each local currency group for multicurrencyportfolios.

The systems and methods described below may apply to any type offinancial products and combinations thereof, including (withoutlimitation): futures, forwards, swaps, ‘vanilla’ options (calls andputs), basic exercise (European and American), options (includingoptions on first line swaps), fixed income products (e.g., swaps (IRS,CDS, Caps, Floors, Swaptions, Forward Starting, etc.)), dividendpayments, exotic options (e.g., Asian Options, Barrier Options,Binaries, Lookbacks, etc.), exercise products (e.g., Bermudan, Canary,Shout, Swing, etc.) and exchange traded derivative (ETD) products. Insome examples, the financial products may include financial productsacross one or more currencies. In some examples, the systems and methodsdescribed below may be applied to ETD products, including ETD interestrate derivative products. An example model system may be configured, insome embodiments, to assess initial margin for cleared interest ratederivative products. Futures products may include, without being limitedto, STIR futures, GCF repo futures, Swapnote® futures, bond futures,Eris futures, EONIA futures and SONIA futures. Options products mayinclude, without being limited to, options on futures (including optionson STIR futures, mid-curve options on STIR futures) and options on bondfutures.

Turning now to FIG. 16, FIG. 16 is a functional block diagramillustrating an exemplary risk management system 1600 (also referred toherein as system 1600). System 1600 may include risk engine 1602, one ormore data sources 1604 and one or more data recipients 1606. Risk engine1602, data source(s) 1604 and data recipient(s) 1606 may becommunicatively coupled via one or more communication networks 1608. Theone or more networks 1608 may include, for example, a private network(e.g., a local area network (LAN), a wide area network (WAN), intranet,etc.) and/or a public network (e.g., the internet).

Each of risk engine 1602, data source(s) 1604, and data recipient(s)1606 may comprise one or more computing devices, including anon-transitory memory storing computer-readable instructions executableby a processing device to perform the functions described herein. Insome examples, risk engine 1602 may be embodied on a single computingdevice. In other examples, risk engine 1602 may refer to two or morecomputing devices distributed over several physical locations, connectedby one or more wired and/or wireless links.

Data source(s) 1604 may include any suitable source of data and/orinformation relating to risk factor data for one or more financialproducts of one or more portfolios and associated with one or morecurrencies. In general, data source(s) 1604 may comprise a servercomputer, a desktop computer, a laptop, a smartphone, tablet, or anyother user device known in the art configured to capture, receive, storeand/or disseminate any suitable data. In one non-limiting example, datasource(s) 1604 include sources of electronic financial data. Data and/orinformation from data source(s) 1604 may include current (e.g.,real-time data) as well as historical data. In some examples, thedata/information may include data for contract and instrumentspecifications for one or more financial products (e.g., ETD products).In some examples, data source(s) 1604 may include one or more electronicexchanges, or any other suitable internal or external data source thatmay either push data to risk engine 1602 or from which risk engine 1602may pull data relating to risk factor data. In general, risk factor datamay be defined as a representation of relevant market risk. Risk factordata may be used for pricing financial instruments in the calculationof, for example, instrument profit and loss, cash rates, forward rateagreements (FRAs), Swaps (e.g., eur otc risk factors, GBP OTC riskfactors, USD OTC risk factors), ETD linear risk factors (e.g., STIRfuture-Eur, GBP, USD, CHF) (quarterly, serial), bond futures, GCF repo,ETD option risk factors-options on STIR futures, mid-curve options onSTIR futures, options on bond futures (e.g., Eur, GBP) and ETD FX riskfactors.

In addition to the risk factor data, data source(s) 1604 may provideadditional data including, without being limited to, trading volume,open interest, bid-ask spread information and spot FX rates. Theadditional information may be used by LRC model 1618 to determine one ormore LRC values.

Data recipient(s) 1606 may include any suitable computer device fordisplaying and/or interacting with risk reporting results that may beprovided by risk engine 1602. Data recipient(s) 1606 may be associatedwith any individual and/or entity for which risk reporting results maybe useful for managing risk associated with one or more financialproducts. In some examples, data recipient(s) 1606 may include userinterface 1634 (e.g., a graphical user interface (GUI)) comprising riskdashboard 1636. Risk dashboard 1636 may be configured to display summaryreports of risk data (e.g., daily risk reports including initialmargin(s) and LRC value(s) associated with one or more portfolios, riskanalytics, other model information). In some examples, user interface1634 (via risk dashboard 1636) may allow a particular data recipient tonot only view the risk data reports, but may also allow the datarecipient to interact with the risk data reports and/or to interact withrisk engine 1602. For example, a specially permissioned administratormay be configured to interact with one or more modules of risk engine(e.g., synthetic data generator 1626, model calibration module 1628,model testing module 1630) to analyze and/or adjust one or moreparameters of model system 1614.

Risk engine architecture 1602 may include one or more data sourceinterfaces 1610, controller 1612, model system 1614, reportgenerator/distributer 1620, one or more data recipient interfaces 1622,one or more databases 1624, synthetic data generator 1626, modelcalibration module 1628 and model testing module 1630. Controller 1612may be specially configured to control operation (via execution ofprogramming logic) of one or more of data source interface(s) 1610,model system 1614, report generator/distributer 1620, data recipientinterface(s) 1622, database(s) 1624, synthetic data generator 1626,model calibration module 1628 and model testing module 1630. Controller1612 may include without being limited to, a processor, amicroprocessor, a central processing unit, an application specificintegrated circuit (ASIC), a field programmable gate array (FPGA), adigital signal processor (DSP) and/or a network processor.

Risk engine 1602 may comprise one or more processors configured toexecute instructions stored in a non-transitory memory (such as shown inFIG. 52). It should be understood that risk engine 1602 refers to acomputing system having sufficient processing and memory capabilities toperform the specialized functions described herein.

Data source interface(s) 1610 may comprise at least one interface (e.g.,an electronic device including hardware circuitry, an application on anelectronic device) for communication with data source(s) 1604. In someexample, data source interface(s) 1610 may be configured to be speciallyformatted for communication with particular data source(s) 1604. In someexamples, data source interface(s) 1610 may be configured to obtain datapushed from among one or more of data source(s) 1604. In some examples,data source interface(s) 1610 may be configured to poll one or more ofdata source(s) 1604 for data, for example, at predefined times,predefined days, predefined time intervals, etc. In some examples, dataobtained via data source interface(s) 1610 may be stored in database(s)1624. In some examples, at least a portion of the obtained data may beprovided to model system 1614.

Model system 1614 may include IRM 1616 and LRC model 1618. In generaleach of IRM 1616 and LRC model 1618 may be comprised as a specializedset of computer-readable instructions (i.e., a set of specialized rules)for respectively determining an initial margin and a LRC value for oneor more portfolios of financial products across one or more currencies.

Referring to FIG. 17, a functional block diagram of example IRM 1616 isshown. IRM 1616 may include volatility forecaster 1702, scenariogenerator 1704, aggregator 1706, base IM generator 1708 and one or moreadditional model components 1710. In general, IRM 1616 may be configuredto receive risk factor data representing market risk for one or moreparticular products. IRM 1616 may leverage appropriate pricingfunctions, to price any contract value starting from a simulated riskfactor time series. More specifically, IRM 1616 may be configured todetermine an exposure which may then be aggregated at the portfoliolevel to produce a portfolio-level initial margin. The portfolio-levelinitial margin may be used to determine a risk exposure.

Volatility forecaster 1702 may be configured to receive risk factordata, including historical data. Based on the risk factor data,volatility forecaster 1702 may be configured to scale historical (raw)risk factor returns by volatility, based on current risk factor values(e.g., risk factor values associated with a current day). Volatilityforecaster 1702 may also be configured to generate one or more riskfactor scenarios based on the volatility-scaled risk factor returns.

Scenario generator 1704 may be configured to receive the risk factorscenario(s) (from volatility forecaster 1702). Based on the risk factorscenario(s), scenario generator 1704 may be configured to generate oneor more price scenarios at the instrument level. In addition, scenariogenerator 1704 may be configured to generate one or more profit and lossscenarios. The profit and loss scenarios may be generated relative to acurrent (e.g., today's) risk factor derived price.

Aggregator 1706 may be configured to receive the individual profit andloss scenarios at the instrument level (from scenario generator 1704).Aggregator 1706 may be configured to aggregate (e.g., combine) theinstrument-level profit and loss scenarios to obtain one or more profitand loss scenarios at the portfolio level.

Base IM generator 1708 may be configured to receive the portfolio-levelprofit and loss scenarios. Base IM generator 1708 may be configured todetermine a base initial margin from a distribution determined from theportfolio-level profit and loss scenario(s), at a predefined percentile.

Additional model component(s) 1710 may include one or more componentsthat may be applied to the base initial margin (determined by base IMgenerator 1708) to obtain a final initial margin. In some examples,additional model component(s) 1710 may include a correlation stresscomponent, a diversification benefits component and a currencyallocation component (each described further below).

IRM 1616 is described further below with respect to FIGS. 20, 21, 22,23A and 23B.

Referring next to FIG. 18, a functional block diagram of example LRCmodel 1618 is shown. LRC model 1618 may include synthetic portfoliogenerator 1802, concentration charge component 1804, bid-ask chargecomponent 1806 and liquidity risk charge component 1808. In general, LRCmodel 1618 may be configured to receive information relating to tradingvolume, open interest, bid-ask spread(s), spot FX rates and one or moreprofit and loss vectors associated with one or more financial products.In some examples, the trading volume information, open interestinformation, bid-ask spread information and spot FX rate information maybe obtained from among data source(s) 1604 (which may be stored indatabase(s) 1624). The profit and loss vector(s) may be obtained fromIRM 1616. The set of received data may be used by LRC model 1618 todetermine any LRC values for one or more portfolios.

Synthetic portfolio generator 1802 may be configured to receive the setof received data (e.g., volume, interest, spread, FX rates and vectors)and may be configured to generate two different representations of aportfolio, based on representative instruments. One of the portfoliorepresentations may be based on a Delta method and the other portfoliorepresentation may be based on a VaR technique. Both representations aredescribed further below.

Concentration charge component 1804 may receive the two portfoliorepresentations from synthetic portfolio generator 1802. Concentrationcharge component 1804 may be configured to determine a concentrationcharge for the concentrated representative instruments for each of thetwo portfolio representations.

Bid-ask charge component 1806 may receive the two portfoliorepresentations from synthetic portfolio generator 1802. Bid-ask chargecomponent 1806 may be configured to determine a bid-ask charge for therepresentative instrument positions for each of the two portfoliorepresentations.

Liquidity risk charge component 1808 may be configured to receive theconcentration charge (from concentration charge component 1804) and thebid-ask charge (from bid-ask charge component 1806). Liquidity riskcharge component 1808 may be configured to determine the LRC value(s)based on a sum of the concentration charge and the bid-ask charge.

LRC model 1618 is described further below with respect to FIGS. 24-26.

Referring back to FIG. 16, report generator/distributer 1620 may beconfigured to receive one or more initial margins (from IRM 1616) andone or more LRC values (from LRC model 1618). In some examples, reportgenerator/distributer 1620 may be configured to receive additionalinformation from among one or more of database(s) 1624, synthetic datagenerator 1626, model calibration module 1628 and model testing module1630. Report generator/distributer 1620 may be configured to create oneor more summary risk reports from among the received information and maydistribute the report(s) to one or more among data recipient(s) 1606(via data recipient interface(s) 1622). The summary risk reports mayinclude, for example, daily risk reports, periodically distributedreports (e.g., according to any predefined time interval) and/or reportsbased on one or more predetermined conditions (e.g., detection of anextreme event). In some examples, the report(s) may include one or moredata analytics associated with one or more portfolios.

In some examples, report generator/distributer 1620 may be configured todistribute report(s) to particular ones of data recipient(s) 1606 atdifferent distribution times (e.g., based on predetermined datarecipient characteristic). In some examples, reportgenerator/distributer 1620 may be configured to distribute reports todata recipient(s) 1606 at a same predetermined time. In some examples,report generator/distributer 1620 may be configured to format the riskreport(s) for presentation on particular ones of data recipient(s) 1606(including, in some examples, differently formatted report(s) forpresentation on different user interfaces 1634 of different datarecipient(s) 1606).

Data recipient interface(s) 1622 may comprise at least one interface(e.g., an electronic device including hardware circuitry, an applicationon an electronic device) for communication with data recipient(s) 1606(e.g., via user interface 1634). In some examples, data recipientinterface(s) 1622 may be configured to be specially formatted forcommunication with particular ones of data recipient(s) 1606. In someexamples, data recipient interface(s) 1622 may be configured to receiveuser input from among data recipient(s) 1606 (e.g., via user interface1634).

Database(s) 1624 may be configured to store any data that may besuitable for use by risk engine 1602. Data stored in database(s) 1624may include, without being limited to, one or more among input data(e.g., risk factor data, volume data, open interest data, spread data,FX rate data) from among data source(s) 1604, one or more values derivedby IRM 1616 and/or LRC model 1618, one or more parameters used by IRM1616 and/or LRC model 1618, information associated with data source(s)1604, information associated with data recipient(s) 1606, synthetic datagenerated by synthetic data generator 1626 and parameters of and/orvalues determined by one or more of model calibration module 1628 andmodel testing module 1630. Database(s) 1624 may be configured accordingto any suitable data structure.

Synthetic data generator 1626 may be configured to create one or moresynthetic data sets. The synthetic data may be configured to modelbenign conditions as well as extreme events. In addition to utilizingrealized market data, risk engine 1602 may be configured to create“synthetic” (i.e. simulated) data, in order to explore model behavior ina controlled environment. With realized data, statistical parametersdescribing the observed distribution must be estimated as the truepopulation generating the data is not known. With synthetic data, thepopulation and its statistical parameters are known as they are directlycreated. In an example embodiment, the synthetic market environmentsgenerated include: benign data and regime change data. The benign datamay be configured to capture the state of the world where markets arewell behaved. This represents a simplified view of the world where noextreme events occur day to day other than what the normal distributionwill allow. Regime change data may be configured to capture the state ofthe world where the market is well behaved initially, then moves intostress abruptly, and then moves back to a well behaved market state.Synthetic data generator 1626 is described further below with respect toFIG. 27.

Model calibration module 1628 may be configured to calibrate one or moreof IRM 1616 and LRC model 1618. In some examples, model calibrationmodule 1628 may be configured to calibrate IRM 1616 and/or LRC model1618 based on real data and synthetic data obtained from synthetic datagenerator 1626. Model calibration module 1628 is described furtherbelow.

Model testing module 1630 may be configured to test IRM 1616 acrossmultiple categories. According to an example embodiment, the testingcategories may include fundamentals testing, backtesting,pro-cyclicality testing, sensitivity testing (including a rollingbacktest analysis and a parameter sensitivity analysis), incrementaltesting, model comparison with historical simulation and assumptionsbacktesting. In some examples, model testing module 1630 may beconfigured to test IRM 1616 based on real data and synthetic dataobtained from synthetic data generator 1626. Model testing module 1630is described further below.

Although not shown, risk engine 1602 may include one or more componentsof risk engine architecture 100 (FIG. 1). For example, risk engine 1602may include one or more of a data filter (to clean erroneous data, fillgaps in data, convert raw data into a time series, etc.), portfoliobucketing and portfolio compression.

Some portions of the above description describe the embodiments in termsof algorithms and symbolic representations of operations on information.These algorithmic descriptions and representations are commonly used bythose skilled in the data processing arts to convey the substance oftheir work effectively to others skilled in the art. These operations,while described functionally, computationally, or logically, areunderstood to be implemented by computer programs or equivalentelectrical circuits, microcode, or the like. Furthermore, it has alsoproven convenient at times, to refer to these arrangements of operationsas modules, without loss of generality. The described operations andtheir associated modules may be embodied in specialized software,firmware, specially-configured hardware or any combinations thereof.

Those skilled in the art will appreciate that risk engine 1602 may beconfigured with more or less modules to conduct the methods describedherein. As illustrated in FIGS. 19-25 and 27, the methods shown may beperformed by processing logic that may comprise hardware (e.g.,circuitry, dedicated logic, programmable logic, microcode, etc.),software (such as instructions run on a processing device), or acombination thereof. In one embodiment, the methods shown in FIGS. 19-25and 27 may be performed by one or more specialized processing componentsassociated with components 1602-1636 of system 1600 of FIG. 1. FIGS.19-25 and 27 are described with respect to FIGS. 16-18. With respect toFIGS. 19-25 and 27, although these flowcharts may illustrate a specificorder of method steps, it is understood that the illustrated methods areexemplary, and that the order of these steps may differ. Also, in someexamples, two or more steps may be performed concurrently or withpartial concurrence,

Referring next to FIG. 19, a flowchart diagram illustrating an examplemethod of operating model system 1614 to determine risk exposure and toadjust one or more parameters of model system 1614 is shown. At step1900, model system 1614 may receive risk factor return data. Asdiscussed above, the risk factor return data may be obtained from amongdata source(s) 1604 and/or via database(s) 1624. In addition to the riskfactor data, model system 1614 may also obtain input data (e.g., tradingvolume information, open interest information, bid-ask spreadinformation and spot FX rate information) for use by LRC model 1618(e.g., from database(s) 1624 and/or data source(s) 1604).

At step 1902, at least one initial margin associated with aportfolio-level profit and loss (PNL) distribution may be determined,for example, by IRM 1616, based on the received risk factor return data.Step 1902 is described further below with respect to FIG. 20.

At step 1904, one or more profit and loss (PNL) vectors may be obtainedby LRC model 1618, from IRM 1616. At step 1906, LRC model 1618 maydetermine one or more LRC values associated with a portfolio-levelliquidity risk, for one or more portfolios. Steps 1904 and 1906 aredescribed further below with respect to FIG. 24.

At step 1908, the initial margin(s) and LRC value(s) may be distributedto one or more among data recipient(s) 1606. For example, reportgenerator/distributer 1620 may generate a risk report based on theinitial margin(s) (determined at step 1902) and the LRC value(s)(determined at step 1906) and may distribute the risk report to datarecipient(s) 1606 via data recipient interface(s) 1622.

At optional step 1910, one or more synthetic datasets may be generated,for example, by synthetic data generator 1626. In some examples, thesynthetic dataset(s) may be generated based at least in part on userinput (e.g., via risk dashboard 1636 of user interface 1634).

At optional step 1912, one or more of IRM 1616 and LRC model 1618 may becalibrated (e.g., various parameters and/or settings of IRM 1616 and/orLRC model 1618 may be set), for example, by model calibration module1628. In some examples, one or more models of model system 1614 may becalibrated using at least one of real data and synthetic data (e.g., asgenerated at step 1910). In some examples, the calibration may beperformed based at least in part on user input (e.g., via risk dashboard1636 of user interface 1634).

At optional step 1914, one or more testing routines may be performed onat least one of IRM 1616 and LRC model 1618, for example, by modeltesting module 1630. In some examples, one or more models of modelsystem 1614 may be tested using at least one of real data and syntheticdata (e.g., as generated at step 1910). In some examples, the testingmay be performed based at least in part on user input (e.g., via riskdashboard 1636 of user interface 1634). Example testing routines aredescribed further below.

At optional step 1916, one or more parameters of IRM 1616 and/or LRCmodel 1618 may be adjusted based on outcome(s) of the testing routine(s)(determined in step 1914). In some examples, the adjustment may beperformed based at least in part on user input (e.g., via risk dashboard1636 of user interface 1634).

Generation of Initial Margin by the Margin Model

Referring next to FIG. 20, a flowchart diagram illustrating an examplemethod of determining at least one initial margin (step 1902 in FIG. 19)is shown. The steps of FIG. 20 may be performed by IRM 1616. At step2000, risk factor (RF) return data may be received by IRM 1616. At step,2002, a volatility forecast may be generated, for example, by volatilityforecaster 1702. In general, a volatility forecast may be generated byscaling the historical (raw) risk factor returns by volatility.Volatility forecasting is described further below.

At step 2004, one or more risk factor scenarios may be generated, forexample, by scenario generator 1704. The risk factor scenarios may bebased on the volatility forecast, and where risk factor scenarios may begenerated based on current risk factor values, for example, by scenariogenerator 1704.

At step 2006, one or more price scenarios at the instrument level may begenerated from the risk factor scenarios, for example, by scenariogenerator 1704. At step 2008, one or more profit and loss (PNL)scenarios at the instrument level may be generated, for example,relative to a current risk factor derived price.

At step 2010, the individual PNL scenario at the instrument level (step2008) may be combined to obtain the profit and loss scenarios at theportfolio level, for example, by aggregator 1706. At step 2012, a baseIM may be determined, for example, by base IM generator 1708, from aportfolio-level PNL distribution at a predefined percentile. ThePortfolio-level PNL distribution may be determined from theportfolio-level PNL scenarios determined at step 2010. At optional step2014, a final IM may be determined, for example, by one or moreadditional components 1710). For example, the final IM may be determinedby applying one or more of a correlation stress component, adiversification benefit component, and a currency allocation component(described further below).

Referring next to FIGS. 21, 22, 23A and 23B, a more detailed examplemethod of determining an initial margin is shown. In particular, FIG. 21is a flowchart diagram illustrating an example method of generating avolatility forecast (step 2002); FIG. 22 is a flowchart diagramillustrating an example method of determining the initial margin basedon the volatility forecast (steps 2004-2014); FIG. 23A is a functionalblock diagram illustrating an example signal flow for determining thevolatility forecast shown in FIG. 21; and FIG. 23B is a functional blockdiagram illustrating an example signal flow for determining the initialmargin shown in FIG. 22. As a note, in some portions of the disclosure,volatility is described as “vol” solely for convenience in notation.

Referring to FIGS. 21 and 23A, generation of the volatility forecast(step 2002) is described. Regarding the volatility forecast (step 2002),the volatility forecast may involve a combination of historicalvolatility process 2100, de-volatilization of risk factor returns, 2102,stress volatility process 2104, APC process 2106, volatility floorprocess 2108, time scaling process 2110 and re-volitization process2112, Processes 2100-2112 are described in greater detail below.

In general, historical volatility process 2100 may include step 2100 aof receiving risk factor time series data for N time series (where N isgreater than or equal to one), step 2100 b of constructing a set ofhistorical EWMA volatilities based on the set of N time series and step2100C of performing de-volitization on the set of historical EWMAvolatilities.

De-volitization of risk factor returns process 2102 may include step2102 a of generating one-dimensional (1D) risk factor returns and step2102 b of performing a de-volitization of the 1D risk factor returns,based on historical volatility(s) (based on step 2100 c).

Stress volatility process 2104 may include step 2104 a of determiningone or more daily changes in the set of historical EWMA volatilities(determined at step 2100 b), step 2104 b of determining a percentile ofdaily change(s) in the set of historical EWMA volatilities (based onstep 2104 a), and step 2104 c of applying a stress volatility treatment.

APC process 2106 may include step 2106 a of determining an RF stressindex and step 2106 b of applying an APC treatment (based on step 2106a).

Volatility floor process 2018 may include step 2108 a of determining apercentile of EWMA volatility, and step 2108 b of applying a volatilityfloor treatment.

Time scaling processing 2110 may include step 2110 a of applying holdingperiod scaling. Re-volitization process 2112 may include step 2112 a ofperforming re-volitization for the risk factor returns (determined atstep 2102 a) based on scaling by volatility.

In general, the volatility forecasting may involve a VaR approach. At ahigh level, a VaR approach involves measuring market risk whereby thereturn of a portfolio is forecasted using probability and statisticalmodels. This type of statistical approach is in contrast to a “scenarioanalysis” approach to measuring market risk, where portfolio returns areforecasted for specific scenarios of input parameters. The advantage ofusing a VaR approach over a scenario analysis approach is that itdescribes a worst loss for a given probability.

The VaR technique includes several advantages. The VaR corresponds to anamount that could be lost with some chosen probability. It measures therisk of the risk factors as well as the risk factor sensitivities. Itcan be compared across different markets and different exposures. It isa universal metric that applies to all activities and to all types ofrisk. It can be measured at any level, from an individual trade orportfolio, up to a single enterprise-wide VaR measure covering all therisks in the firm as a whole. When aggregated (to find the total VaR oflarger and larger portfolios) or disaggregated (to isolate componentrisks corresponding to different types of risk factor) it takes accountof dependencies between the constituent assets or portfolios.

In general, the VaR may be determined by: specifying a confidence levelq or alternatively the significance level (1−q), specifying a holdingperiod h (also referred to as the risk horizon, or the Margin Period ofRisk (MPOR)) in business days and identifying the probabilitydistribution of the profit and loss (P&L) of the portfolio.

For IRM 1616. a full valuation approach is selected, meaning portfoliosmay be priced for the scenarios under consideration, rather than using alocal valuation approach that approximates the change in the portfoliovalue using local sensitivities. A primarily non-parametric approach isalso selected, meaning that the P&L distribution is constructed based onempirical data, rather than making a distribution assumption. Ingeneral, the choice of a full valuation approach may be preferredbecause approaches like the delta-normal approach capture linear riskexposures only. In addition, both the full valuation approach andnon-parametric approach are feasible because a substantial risk factorhistory is stored from which to derive scenarios.

The particular resolution method chosen for IRM 1616 is a filteredHistorical Simulation (FHS) approach. This approach allows for themodeling of these stylized properties. In describing the FHS resolutionmethod, the Historical Simulation (HS) approaches are examined first andthen it is distinguished how FHS improves on HS. How IRM 1616 is aspecific variation of FHS is then shown.

For implementing historical simulation, consider a length of a lookbackwindow, historical weighting for sampling, sampling scheme and any timescaling or time aggregation. The length of the lookback window refers tohow many historical returns are used looking backward from the valuedate T in the construction of the P&L distribution. In general, theerror of the VaR estimator decreases as the number of availableobservations increases. For example, computation of the 1% VaR using adistribution with only 100 empirical observations is subject to moreestimation error than if 1,000 observations were available. For IRM1616, the lookback window was chosen, in one example, to be 1,250. Thepotential drawback of a longer lookback window is that returns far backin the past have equal influence at the percentile calculation stepcompared to more recent observations.

A potential way to remedy this drawback is through the choice ofhistorical weighting. A choice for equal weighting assigns the sameprobability to each past observation. For IRM 1616, equal historicalweighting is selected, but note that FHS provides an alternate way toremedy the drawback of past history not representing more recent marketconditions.

In an example embodiment, IRM 1616 uses a sampling without replacementtechnique. Sampling without replacement represents a straightforwardapproach, which builds the empirical distribution directly from thehistorical observations in a one-to-one manner. That is, if the lookbackwindow contains 1,000 return observations, then the simulation resultsin an empirical P&L distribution containing 1,000 observations exactlyreflecting each historical return observation.

For sampling without replacement, time scaling may be accomplished byestimating a 1-day VaR and scaling it up to the required h-day timehorizon. This is due to limited data availability since estimating amultiday time horizon VaR with multiday returns typically requires alonger history. For IRM 1616, an option for time scaling is to scale a1-day VaR up to a desired h-day time horizon.

One methodology for time scaling the VaR estimate, which essentiallyrepresents a quantile of an empirical return distribution, is using thesquare-root-of-time rule. For IRM 1616, the square-root-of-time rule isused only to scale the one-day volatility forecast to an h-dayvolatility forecast. For this case, the only assumptions made are thatthe one-day returns are uncorrelated and have constant variance over theholding period.

An enhancement to HS is Filtered Historical Simulation. FHS may bedistinguished from HS by defining the term filtered to mean that ratherthan sampling from a set of raw returns, sampling is done from a set ofshocks or innovations, z_(t), that are filtered by a conditionalvolatility:

$\begin{matrix}{z_{t} = \frac{r_{t}}{\sigma_{{co{nd}};t}}} & (69)\end{matrix}$

The conditional volatility σ_(cond; t) can be estimated using variousmodels, including the Exponentially Weighted Moving Average and theGeneralized Auto-Regressive Conditional Heteroskedastic models. Notethat the filtering volatility is the current estimated conditionalvolatility at the time of the return. To account for current marketconditions the shocks are multiplied by the one-day ahead (T+1)volatility forecast estimated on the value date T. Thus a set of scaledreturns {tilde over (r)}_(t) would be constructed as follows:

$\begin{matrix}{{\overset{˜}{r}}_{t} = {\frac{\sigma_{{cond};{T + 1}}}{\sigma_{{cond};t}}r_{t}}} & (70)\end{matrix}$

Note that this approach can be used for sampling with or withoutreplacement. In the case of sampling with replacement, multiday returnscan be generated using the sampled shocks and a modeled conditionalvolatility/mean updated at each time step. For sampling withoutreplacement, the scaled returns are used to calculate the VaR directlyby applying them to a portfolio on the value date T and calculating aP&L distribution. This filtering and forecasting process is referred tomore generally as applying a volatility adjustment. The filtering andforecasting processes are also referred to as de-volatilization andre-volatilization. In IRM 1616, the EWMA conditional volatility is usedfor filtering returns at each time t. For forecasting, the EWMAconditional volatility is combined with a Stress Volatility Componentand Volatility Floor to derive a T+1 volatility forecast.

Advantages of FHS are similar to HS, but with the additional advantagesthat it adjusts for the current market conditions and allows for aconditional volatility model which can account for heteroskedasticity ofvolatility (i.e. non-constant variance through time).

In summary, the VaR resolution method chosen for IRM 1616 includes thefollowing basic features: historical simulation via equally weightedsampling without replacement; filtering via EWMA conditional volatility,volatility forecasting via EWMA conditional volatility, floor, andstress components; and time scaling of the volatility forecast via thesquare-root-of-time rule.

Indexing

The scaling operations described below are performed for all of the f=1,2, . . . , F risk factors unless otherwise noted, with F denoting thetotal number of Risk Factors including foreign exchange (FX).

Indexing across time is somewhat more involved, but essentially involvesfive primary dates. These dates are:

-   -   The First Date (FD): the first date with Risk Factor return data        available in history.    -   The Last Date (LD): the last date with Risk Factor return data        available in history.    -   The First Value Date (FVD): the first date within the range {FD,        . . . , LD} for which sufficient Risk Factor return data exist        to compute an initial margin given a chosen VaR lookback window        length.    -   The First Backtest Date (FBD): the first date within the range        {FVD, . . . , LD} for which an initial margin calculation for        backtesting purposes is performed.    -   The current Value Date (VD): a date in the range {FVD, . . . ,        LD} that represents the current calculation date for initial        margin.

Importantly, the historical time period covered is identical across allRisk Factors in scope, such that any one row vector chosen from the datamatrix represents the same date for all Risk Factors.

Historical indexing of time is defined with t=1, . . . , T_(max), wheret=1 is the first date available in history (FD) and t=T_(max) is thelast date available in history (LD). The current Value Date is denotedas t=T.

For operations with returns within the lookback window τ=1, . . . , Ware used. For dates where initial margin is calculated, W=W₀ where W₀ isthe default VaR lookback window. For calculating EWMA volatilityestimates on a given Value Date, W can be smaller than W₀ for a dateprior to the FVD; in these cases W=t.

For initial margin calculation, W₀ Risk Factor scenarios may be based onW₀ daily Risk Factor returns. As such, the FVD is the first date t whereW₀+1 daily Risk Factor returns are available, i.e. at time t=W₀+1.

For operations involving instruments, the instruments are indexed byi=1, 2, . . . , N where N is the total number of instruments in a givenportfolio.

Risk Factor Return Time Series Generation

For each Risk Factor f at time t, the daily return is calculated. ForRisk Factors which are interest rates (including cash rates, swaps,FRAs, STIR futures, Repo futures, and bond futures), what are generallyknown as one-day “absolute returns” are calculated, that is, one-daychanges in yield levels y_(t) so that

$\begin{matrix}{r_{t} = {y_{t} - y_{t - 1}}} & (71)\end{matrix}$

At step 1208, message delay line system (e.g., message departure timemodule 212) may determine the TCC propagation delay by comparing thelocal timestamp (applied to the received message by timestamp module 210via wall clock 204) to the remote timestamp included in the receivedmessage (applied by active CIP component 808). In other words, adifference between the local timestamp and the remote timestamp may beused to measure the TCC propagation delay for the message associatedwith a participant. Message departure time module 212 may store the TCCpropagation delay in the participant record (in storage 218) as theparticipant delay offset.

For option volatility Risk Factors and FX Risk Factors, what are knownas logarithmic, or log, returns are calculated as:

$\begin{matrix}{r_{t} = {{\ln\left( \frac{y_{t}}{y_{t - 1}} \right)} = {{\ln\; y_{t}} - {\ln\; y_{t - 1}}}}} & (72)\end{matrix}$

as both volatility and FX are defined to be positive. To lighten thenotation, any script references to risk factor f are omitted.

The return at position τ in the lookback window for historical date t isdefined as:

$\begin{matrix}{{r_{\tau,t} = {{r_{t - W + \tau}\mspace{14mu}{\forall\tau}} = 1}},{.\;.\;.}\;,W} & (73)\end{matrix}$

De-Volatilization of Risk Factor Returns

The first step in the de-volatilization (DV) process is to obtain a setof historical next-day variance predictions {σ_(DV; t) ²}. For datest=1, 2 . . . , M, the historical next-day variance predictions arecalculated as the sample variance (ignoring the sample mean) based onthe first M dates:

$\begin{matrix}{\sigma_{{DV};t}^{2} = {\frac{1}{M}{\sum\limits_{\tau = 1}^{M}r_{\tau}^{2}}}} & (74)\end{matrix}$

For each of the historical dates t=M+1, . . . , T_(max) an EWMA varianceupdating process is used to perform EWMA de-volatilization. The initialseed for the EWMA de-volatilization on date t is set to be the samplevariance (ignoring the sample mean) based on the lookback window W:

$\begin{matrix}{\sigma_{{\tau = 1},t}^{2} = {\frac{1}{W}{\sum\limits_{\tau = 1}^{W}r_{t - W + \tau}^{2}}}} & (75)\end{matrix}$

For calculation of volatility estimates when t<W₀, W=t is set so thatthe EWMA process is run over a time period with less than W₀ returns.The EWMA variance updating process is written in its standard recursiveform as

$\begin{matrix}{{\sigma_{\tau,t}^{2} = {{\lambda \cdot \sigma_{{\tau - 1},t}^{2}} + {\left( {1 - \lambda} \right) \cdot r_{{\tau - 1},t}^{2}}}},} & (76)\end{matrix}$

where τ=2, . . . , W+1 and where 0≤λ<1 is the EWMA Weighting (Lambda)parameter that assigns more weight to more recent observations (when itis strictly less than 1). The value of the EWMA variance σ_(τ,t) ² withτ=W+1 is used to calculate the next-day variance prediction σ_(DV; t) ²for historical date t:

$\begin{matrix}{\sigma_{{DV};t}^{2} = \sigma_{{W + 1},t}^{2}} & (77)\end{matrix}$

For a given Value Date T σ_(DV; t) ² is computed for each t=T−W, T−W+1,. . . ,T−1. Thus the set {σ_(DV; t) ²} of historical next-day variancepredictions and the corresponding set {σ_(DV; t)} of historical next-dayEWMA-DV volatilities are obtained.

As can be seen from the EWMA variance equation (76), the prediction forthe next-day variance is a weighted average of two terms:

-   -   1. The Persistence term: This first term captures how much of        the prior period volatility forecast should be used in the        forecast for the next period. As an illustration, if 100% of the        weight is assigned to this term (λ=1), the forecast volatility        will remain static (at its prior level) regardless of the        volatility experienced in the market today. In this case, if        yesterday's forecast was for low volatility, and the market        experienced high volatility today, the forecast would remain for        low volatility.    -   2. The Reaction term: This second term captures how much current        market volatility, as measured by the squared day-over-day        return, should affect the volatility forecast. As an        illustration, if 100% of the weight is assigned to this term        (1−λ=1 so that λ=0), then the volatility forecast for tomorrow        is always equal to today's market volatility. The lower the        value of Lambda, the more reactive to today's market the        volatility forecast becomes. This term serves as the updating        term to incorporate a current view of volatility.

Both terms capture important features of the data, and neitherextreme—Lambda zero or one—is typically justified in the interest ratesmarket. A mixture of both is desirable.

Finally, each of the returns in the lookback window are scaled:

$\begin{matrix}{{r_{{{DV};\tau},T} = {\frac{r_{\tau,T}}{\sigma_{{DV};{T - W + \tau - 1}}} = \frac{r_{T - W + \tau}}{\sigma_{{DV};{T - W + \tau - 1}}}}},{{{for}\mspace{14mu}\tau} = 1},2,\ldots\mspace{14mu},W} & (78)\end{matrix}$

Re-Volatilization of Risk Factor Returns

The re-volatilization (RV) process applies the most current volatilityestimate to each Risk Factor by scaling the returns using the VolatilityForecast. Unlike in the DV process, the RV forecast volatility is aconstant that applies across the VaR window for a given Risk Factor f.

The RV process consists of the following three steps:

-   -   1) EWMA Volatility estimation for each Value Date    -   2) Forecast Volatility calculation, including calculation of the        Volatility Floor, Stress Volatility Component and APC Treatment    -   3) Risk Factor return scaling using the Volatility Forecast.

EWMA Volatility Estimation

On any given Value Date t=T the EWMA-RV volatility estimate is thesquare root of the EWMA-DV variance estimate at time t=T:

$\begin{matrix}{\sigma_{{RV};T} = {\sqrt{\sigma_{{DV};T}^{2}} = \sigma_{{DV};T}}} & (79)\end{matrix}$

where σ_(DV; T) is obtained using the EWMA process leading to equation(77) with t=T. The above EWMA volatility estimates are used in thepreviously described DV calculations as well as the subsequentcalculations of the Volatility Floor and the Stress VolatilityComponent.

Volatility Floor

The volatility of the previous section is the margin model's predictionof future volatility to be realized over the holding period. To ensureoverall a conservative margin estimate in a low market volatilityenvironment, the forecast volatility is evaluated against a volatilityfloor on each value date.

The volatility floor σ_(Floor) places a minimum value on the forecastvolatility of each risk factor. In an effort to accommodate the uniquevolatility profiles of the risk factors, IRM 1616 provides each riskfactor with its own volatility floor. The volatility floor is alsoanchored historically in that it always includes volatility estimatesreaching back to the first date. In this way the volatility floor isalso a function of time: as volatility increases (decreases), thevolatility floor adapts to the current market environment by increasing(decreasing).

For a given value date T, the minimum is set so that the RV volatilitynever falls below a designated Volatility Floor percentile, k_(floor),of the volatility estimates starting with the first estimate date wheret=1 and moving up to the Value Date t=T:

$\begin{matrix}{{\sigma_{{Floor};T} = {{percentile}\left( {\left\{ \sigma_{{RV};t} \right\}_{1 \leq t \leq T},\ k_{floor}} \right)}},{T \geq {M + K}}} & (80)\end{matrix}$

Here K is a configurable parameter determining the minimum number ofvalues needed for the percentile calculation (in one example, K is setat 750). Note that, in this example, percentile corresponds to theMicrosoft Excel® (function=PERCENTILE.INC( ) with the percentile in thiscase set to k_(floor).

To clarify, the volatility floor calculation is best illustrated with anexample; for the purpose of illustration only the k_(floor) percentileis set to 10%:

-   -   On the First Value Date there exist 1,251 individual σ_(RV; T)        estimates per Risk Factor. The 1,251 Value Date volatility        estimates are ranked from smallest to largest, and the        k*(n−1)+1=126^(th) “observation” (rank) is chosen as the        Volatility Floor for this First Value Date. The rank value in        this case is calculated as the 126^(th) smallest volatility.    -   On an arbitrary Value Date, Mar. 31, 2015, there exist 3,227        individual σ_(RV; T) estimates per risk factor. The Volatility        Floor is then calculated as the k*(n−1)+1=323.6^(th)        “observation” (rank), which is equal to the 323^(rd) smallest        sigma plus 0.6 times the difference between the 324^(th) and        323^(rd) smallest RV volatility.

Stress Volatility Component

IRM 1616 may be configured to ensure that, for the purpose ofdetermining volatilities, periods of market stress are included.

The series of daily changes in the EWMA volatility estimates on eachValue Date is defined as dσ_(RV; t)=σ_(RV; t)−σ_(RV; t-1). Thepercentile of these daily changes is computed to obtain:

$\begin{matrix}{{{\Delta\sigma_{{RV};T}} = {{percentile}\left( {\left\{ {d\;\sigma_{{RV};t}} \right\}_{2 \leq t \leq T},\ k_{delta}} \right)}},{T \geq {M + K}}} & (81)\end{matrix}$

where k_(delta) is the Stress Volatility Update Percentile set to alevel considered as stressed in financial markets. The term accounts forlarge daily changes in volatility.

The Stress Volatility Component for each risk factor f is calculated asfollows:

$\begin{matrix}{{\Delta\sigma}_{{stress};T} = {\alpha_{{APC};T}*{\Delta\sigma}_{{RV};T}}} & (82)\end{matrix}$

where α_(APC; T) is the RF APC Index calculated as:

$\begin{matrix}{\alpha_{{APC};T} = \left\{ {{\begin{matrix}0 & {{{for}\mspace{14mu}\sigma_{{RV};T}} \geq \sigma_{\max;T}} \\\frac{\sigma_{\max;T} - \sigma_{{RV};T}}{\sigma_{\max;T} - \sigma_{{APC};T}} & {{{for}\mspace{14mu}\sigma_{{APC};T}} < \sigma_{{RV};T} < \sigma_{\max;T}} \\1 & {{{for}\mspace{14mu}\sigma_{{RV};T}} \leq \sigma_{{APC};T}}\end{matrix}\sigma_{\max;T}} = {\max\left( {\left\{ \sigma_{{RV};t} \right\}_{1 \leq t \leq T},{{T \geq {M + {K\sigma_{{APC};T}}}} = {{{{percentile}\left( {\left\{ \sigma_{{RV};t} \right\}_{1 \leq t \leq T},k_{APC}} \right)}T} \geq {M + K}}}} \right.}} \right.} & (83)\end{matrix}$

where k_(APC) is the APC Index Percentile and whereσ_(max; T)>σ_(APC; T) at any T.

The RF APC Index α_(APC; T) acts in an anti-procyclical manner at theRisk Factor level, as can be seen in the definitions of the RF APC Indexabove: as the estimated volatility σ_(RV; T) increases, the RF APC Indexapproaches and eventually reaches zero, so that the Stress VolatilityComponent also approaches and eventually reaches zero. At the same time,during periods of low volatility (up to the volatility thresholdσ_(APC; T)), α_(APC; T) evaluates to 1 so that the full term Δσ_(RV; T)applies.

It should be noted that IRM 1616 may not only rely on the StressVolatility Component for its APC performance. Anti-procyclicality ofinitial margin is further achieved by implementing “Option (a)” of theEMIR RTS. This is discussed next.

Treatment of Procyclicality and Volatility Forecast

The Volatility Forecast in IRM 1616 is generated using one of the twoconfigurations: APC Off and APC On. The Volatility Forecast in the APCOn configuration is used to generate an initial margin that complieswith the applicable anti-procyclicality regulatory requirements and thatis used in production. The initial margin generated based on the APC OffVolatility Forecast may be used for testing purposes.

In the case of APC Off, the Volatility Forecast {circumflex over(σ)}_(T) on the Value Date T is calculated by applying the VolatilityFloor and the Stress Volatility Component to the EWMA-RV volatilityσ_(RV; T) as follows:

$\begin{matrix}{{\hat{\sigma}}_{T} = {{\min\left( {\sigma_{\max;T},{\max\left( {{\sigma_{{RV};T} = {\Delta\sigma}_{{stress};T}},\sigma_{{Floor};T}} \right)}} \right)} = {\min\left( {\sigma_{\max;T},{\max\left( {{\sigma_{{RV};T} + {\alpha_{{APC};T}*{\Delta\sigma}_{{RV};T}}},\sigma_{{Floor};T}} \right)}} \right)}}} & (84)\end{matrix}$

Note that the Volatility Forecast is capped at the maximum historicalvolatility σ_(max; T) to avoid a situation where margin may be reducedas market conditions are deteriorating.

The APC treatment in IRM 1616 is implemented by applying a 25% minimumbuffer at the risk factor level. Accordingly in the APC On configurationthe Volatility Forecast {circumflex over (σ)}_(T) is calculated asfollows:

$\begin{matrix}{{\hat{\sigma}}_{T} = {{\min\left( {\sigma_{\max;T},{\max\left( {{\sigma_{{RC};T} + {\max\left( {{\Delta\sigma}_{{stress};T},{\alpha_{{ACP};T} \cdot {APC}_{buffer} \cdot \sigma_{{RV};T}}} \right)}},\sigma_{{Floor};T}} \right)}} \right)} = {\min\left( {\sigma_{\max;T},{\max\left( {{\sigma_{{RV};T} + {\alpha_{{APC};T} \cdot {\max\left( {{\Delta\sigma}_{{RV};T},{{APC}_{buffer} \cdot \sigma_{{RV};T}}} \right)}}},\sigma_{{Floor};T}} \right)}} \right)}}} & (85)\end{matrix}$

where APC_(buffer) is set to a minimum value of 25%. The termα_(APC; T)·APC_(buffer)·σ_(RV; T) plays the role of a buffer similar tothat of the Stress Volatility Component Δσ_(stress; T). Accordingly, themaximum of these terms is taken in equation (85).

The election of the minimum buffer value (e.g., 25%) ensures that abuffer is applied at the Risk Factor level during periods of marketcalm, so that increases in the Volatility Forecast during periods ofmarket stress can be more gradual as the buffer is designed to beexhausted as market stress (volatility) increases.

Scaling Risk Factor Returns by Volatility

As a final step in the RV scaling process, the raw returns are scaled bythe ratio of the Volatility Forecast {circumflex over (σ)}_(T) to the DVvolatility:

$\begin{matrix}{{r_{{{RV};\tau},T} = {{{\hat{\sigma}}_{T} \cdot r_{{{DV};\tau},T}} = {\frac{{\hat{\sigma}}_{T}}{\sigma_{{DV};{T - W + \tau - 1}}}r_{T - W + \tau}}}},\ {\forall{\tau \in \left\lbrack {1,W} \right\rbrack}}} & (86)\end{matrix}$

This return scaling step applies to all Risk Factors, including, forexample, linear Risk Factors, option Risk Factors, and FX Risk Factors.At this point there exist W₀ Risk Factor return scenarios, obtained byindependently scaling the raw returns by their respective ratios of theVolatility Forecast to the DV volatility. The RF return scenarios aregenerated either in APC Off or APC On configuration by using equation(84) or equation (85) for Volatility Forecast {circumflex over (σ)}_(T),respectively.

Example of Generating Scaled Return Scenarios

An example of the scaling calculations up to this point are provided tohighlight the fact that IRM 1616 runs a separate EWMA process for eachdate within the lookback window. In this manner, IRM 1616 estimatesvariances consistently with the same process for each day, with each dayin the lookback period using, for example, 1250 observations (subject todata availability in the early years).

An arbitrary Value Date of Aug. 12, 2008 is selected, which correspondsto T=1515 within the historical dataset. The process of scaling a singlereturn at time t=1505 is illustrated, which is one of the 1250 returnswithin the lookback period for T=1515. Each of the days within a givenlookback window are indexed by τ from 1 to W, so that τ=1250 correspondsto the Value Date and τ=1240 corresponds to the single return to beillustrated: r_(τ,t)=r_(1240,1515). Here the time indexing again showsthat the 1240^(th) return targeted within the lookback period for thehistorical day T=1515.

For simplicity the illustration does not include computation of theStress Volatility Component, the Volatility Floor or APC treatment.

The example process is as follows:

-   -   1. Unscaled Returns: Compute 1-day returns r_(t) for t=1, 2, . .        . , 1515 using equation (71) or (72).    -   2. Devolatilization: Generate the DV volatility to be applied to        the return r_(τ,t).        -   a. Compute the seed value σ_(τ=1,t) ² using equation (75)            with W=1250 and t=1504.        -   b. Run the EWMA variance updating scheme for τ=2, . . . ,            W+1 using equation (76).        -   c. Obtain the next-day variance prediction σ_(DV,t) ² using            equation (77) with t=1504.        -   d. Obtain σ_(DV,t) by taking the square root of σ_(DV,t) ²            with t=1504.    -   3. Volatility Forecasting: Generate the volatility forecast for        T=1515:        -   a. Run the EWMA variance updating scheme for τ=2, . . . ,            W+1 using equation (76) to obtain the next-day variance            prediction σ_(DV,t) ² for t=1515 using equation (77).        -   b. Compute σ_(RV; T) using equation (79) with T=1515 and            take it as the Volatility Forecast {circumflex over            (σ)}_(T). Note that the Volatility Forecast is the same for            any time τ return within the lookback period.    -   4. Scaling Risk Factor Return by Volatility: Obtain a scaled        return for t=T−W+τ=1505 using equation (86) with τ=1240, W=1250,        T=1515. Note that T−W+τ−1=1504, and that σ_(DV; T−W+τ−1) was        computed in Step 2d.    -   The four step process above is performed for each τ=1, . . .        , W. Thus W scaled returns are obtained for a given Value Date        T.

Accommodating Different Holding Periods

The returns are calculated as daily price changes (either absolutereturns or log returns). To accommodate different holding periods, the1-day holding period Volatility Forecast is scaled by thesquare-root-of-time rule:

$\begin{matrix}{{\hat{\sigma}}_{h;T} = {{\hat{\sigma}}_{{h = 1};T} \cdot \sqrt{h}}} & (87)\end{matrix}$

The time-scaling of volatility is applied at the RV stage detailedabove. Therefore the h-day holding period equivalent of the EWMA-RVreturn may be re-written as:

$\begin{matrix}{r_{{RV},{h;\tau},T} = {\sqrt{h} \cdot r_{{{RV};\tau},T}}} & (88)\end{matrix}$

All remaining calculations above continue to be conducted on 1-dayreturns as previously stated.

Next, referring to FIGS. 22 and 23B, determination of at least oneinitial margin based on the volatility forecasting (steps 2004-2014) isdescribed. Regarding determination of the initial margin(s), thedetermination may involve a combination of profit and loss (PNL) process2200, correlation stress process 2202, diversification benefit process2204 and currency allocation process 2206. Processes 2200-2206 aredescribed in greater detail below.

In general, profit and loss process 2200 may include step 2200 a ofgenerating one or more risk factor scenarios, step 2200 b of generatingone or more instrument pricing scenarios, step 2200 c of generatinginstrument rounded profit and loss scenarios, step 2200 d of aggregatingthe instrument rounded profit and loss scenarios to generate one or moreportfolio-level profit and loss scenarios, step 2200 e of determining atleast one base initial margin from a portfolio-level VaR, step 2200 f ofgenerating one or more sector-based sub-portfolio-level profit and lossscenarios and step 2200 g of generating one or more currency-basedsub-portfolio-level profit and loss scenarios for one or morecurrencies.

Correlation stress process 2202 may include step 2202 a of determiningindividual instrument-level VaR and step 2202 b of applying acorrelation stress component to the base initial margin(s) (step 2200e).

Diversification benefit process 2204 may include step 2204 a ofdetermining sector-level VaR (based on step 2200 f) and step 2204 b ofapplying a diversification benefit to the base initial margin(s) (step2200 e) based on the sector-level VaR (step 2204 a).

Currency allocation process 2206 may include step 2206 a of determiningcurrency-based VaR (based on step 2200 g), step 2206 b of applying thecurrency-based VaR to the initial margin (after applying thediversification benefit of step 2204 b) and step 2206 c of allocatingthe initial margin (after step 2206 b) among one or more currencies toform the final initial margin.

Risk Factor Scenarios

A Risk Factor Scenario refers to the application of RF returns to therisk factor level on a given Value Date. The purpose of this process isto create a distribution around the RF level that is consistent with theEWMA volatility scaling of returns discussed above.

For Risk Factors using absolute returns (i.e. linear Risk Factors), theh-day scenarios for a given Risk Factor f are constructed by adding theh-day Risk Factor returns to the Risk Factor level y_(T) on the ValueDate T as follows:

$\begin{matrix}{{{RF}\mspace{14mu}{Scenario}_{\tau}} = {= {y_{T} + r_{{RV},{h;\tau},T}}}} & (89)\end{matrix}$

For Risk Factors using log returns (i.e. option Risk Factors and FXrates), the h-day scenarios for a given Risk Factor f are constructed byapplying the h-day Risk Factor returns to the Risk Factor level y_(T) onthe Value Date as follows:

$\begin{matrix}{{{RF}\mspace{14mu}{Scenario}_{\tau}} = {= {y_{T} \cdot e^{r_{{RV},{h;\tau},T}}}}} & (90)\end{matrix}$

No pricing occurs at this stage. The RF scenarios are generated eitherin APC On or APC Off configuration using equation (84) or equation (85)for Volatility Forecast {circumflex over (σ)}_(T), respectively({circumflex over (σ)}_(T) is used in the calculation of the RF returnscenarios r_(RV,h; τ,T)).

Instrument-Level Price Scenarios

This step in the margin model process uses pricing functions on the RiskFactor Scenarios to calculate base prices at the instrument level aswell as price scenarios. Pricing functions are denoted by PF to provideonly a high level overview of the basic pricing steps.

Regarding base prices: for instrument i the base price on Value DateVD=T is calculated as:

$\begin{matrix}{p_{i,T} = {P{F\left( {\left\{ y_{T} \right\},T} \right)}}} & (91)\end{matrix}$

The Risk Factor level y_(T) is in constant time to maturity space withrespect to Pricing Date (the Value Date T in this case). Accordingly aninterpolation on the {y_(T)} curve occurs in the pricing function PF tocalculate the RF level corresponding to the time to expiry.

Regarding price scenarios: for an instrument i the set of pricescenarios indexed by τ on the Value Date T is calculated as:

$\begin{matrix}{= {P{F\left( {{\{\}},{VD}_{h}} \right)}}} & (92)\end{matrix}$

Here VD_(h) is the business date after the Value Date such that thenumber of business days between VD and VD_(h) is equal to h (the holdingperiod) if the instrument expiry is at the end of the holding period orlater. For instruments expiring within the holding period, VD_(h) is setto the expiry date of the instrument. An interpolation in the pricingfunction PF is performed here at the RF scenario level

.

An exception to the above base prices and price scenarios may apply tobond futures: bond future implied yields are used instead of bond yieldsderived from the Risk Factors as pricing based on the latter may resultin a mismatch with settlement prices. Specifically, the base price iscalculated as:

$\begin{matrix}{p_{i,T} = {P{F\left( {\left\{ y_{i,T}^{S} \right\},T} \right)}}} & (93)\end{matrix}$

where y_(i,T) ^(S) is the Bond Future Implied Yield that results inp_(i,T) being the end-of-day settlement price of the bond futuresinstrument on the Value Date T.

The set of price scenarios for bond futures indexed by τ is calculatedas:

$\begin{matrix}{= {P{F\left( {\left\{ {y_{i,T}^{S} + {\hat{r}}_{{RV},{h;\tau},T}} \right\},{VD}_{h}} \right)}}} & (94)\end{matrix}$

Here {circumflex over (r)}_(RV,h; τ,T) is the yield return scenarioobtained by linear interpolation (with flat extrapolation) on thegovernment bond yield RF returns {r_(RV,h; τ,T)} (e.g., with maturities2Y, 5Y, 10Y, 30Y), corresponding to the instrument's time to expiry.Only the RF scenario returns {r_(RV,h; τ,T)} are interpolated in thiscase since y_(i,T) ^(S) is already at the Expiry Date of the instrument.

It should be noted that for scenarios all instruments are priced onValue Date VD_(h): the values for each Risk Factor scenario and the dateVD_(h) are inputs to the pricing function so that the output is theinstrument price for the given scenario. Since the future realizedfixings are not available until the date VD_(h), implied fixingscomputed based on the yield curve (constructed on the Value Date VD) areused as the corresponding forward rates.

For each instrument there now exist W=W₀ scenarios in price space. NoP&L is calculated at this stage, and no CVF or contract count isattached.

Instrument-Level P&L Scenarios

This step in the margin model process calculates the Profit & Lossscenario for each instrument i separately. Let

be the price scenario and let p_(i,T) be the base price on Value Date Tfor instrument i. The P&L for a given instrument i and scenario τ isthen calculated as follows:

$\begin{matrix}{{PnL_{i,\tau}} = {\left( {- p_{i,T}} \right)*CVF_{i}}} & (95)\end{matrix}$

where CVF_(i) is the Contract Value Factor specific to instrument i.Only the P&L at the individual instrument level exists, including itsCVF. The position size is not included at this stage.

In the subsequent Sections, bold symbols to refer to a vector ofscenario values indexed by τ, e.g.:

$\begin{matrix}{{PnL_{i}} = \begin{bmatrix}{PnL_{i,{\tau = W}}} \\\ldots \\\ldots \\{PnL_{i,{\tau = 1}}}\end{bmatrix}} & (96)\end{matrix}$

P&L Rounding at the Instrument Level

IRM 1616 may be configured to generate high levels of decimal accuracy.This is convenient for the purpose of calculating a large number ofportfolio values to a high degree of accuracy. However, financialmarkets typically quote the prices of financial instruments to a muchlower resolution (i.e. to fewer decimal points).

To ensure that Variation Margin (VM), due to reported settlement pricechanges, and instrument level P&L, due to Model price changes, areconsistent with each other, the P&L at the instrument level may berounded to the nearest minimum increment:

$\begin{matrix}{{PnL_{{{{roun}d};i},\tau}} = {\left\lceil \frac{PnL_{i,\tau}}{\Delta_{i}} \right\rceil*\Delta_{i}}} & (97)\end{matrix}$

where P&L_(i,τ) is the Profit and Loss vector for instrument i atscenario τ, Δ_(i) is the instrument-specific minimal increment, andwhere the bracket operator ┌⋅┐ represents the function that rounds upthe value away from zero to the nearest integer (corresponding to=ROUNDUP(⋅,0) in Microsoft Excel®) with one exception. In some examples,if the absolute value within the square brackets is less than 1/100,i.e. P&L_(i,τ), is within 1/100^(th) of a tick size of zero, then thefunction will round the value to zero. Note that Δ=CVF*ticksize.

Portfolio-Level P&L Scenarios and Base IM

For single currency portfolios, the Portfolio Level P&L vector iscalculated by summing the Instrument Level P&L vectors (multiplied bytheir respective quantities):

$\begin{matrix}{{PnL_{\prod}} = {\sum\limits_{i}{Q_{i} \cdot {PnL}_{i;\prod}}}} & (98)\end{matrix}$

where the summation is across all instruments in portfolio Π, and whereQ_(i) is the quantity (position) held of instrument i (which can bepositive or negative to indicate a long or short position). Thisproduces a single distribution for the P&L of a given portfolio.

Given a VaR confidence level q, the Base IM for portfolio Π is obtainedby taking the percentile 1−q of the P&L distribution:

$\begin{matrix}{{I\; M_{\Pi}} = {\min\left( {{{percentile}\left( {{{Pn}L_{\Pi}},{1 - q}} \right)},0} \right)}} & (99)\end{matrix}$

Note that percentile corresponds to the Microsoft Excel®function=PERCENTILE.INC( ). The Initial Margin IM_(Π) may be capped atzero as margin is concerned with collateralizing potential losses,rather than gains.

Multi-Currency Portfolio Base IM via Currency Allocation Component

For single currency portfolios the model steps are as described above.No FX conversion takes place. Multi-currency portfolios receive theadditional steps to convert from the various local currencies (thecurrencies in which any given trades are settled) to the IM BaseCurrency (the single currency in which Initial Margin is calculated),which may be handled by a currency allocation component (e.g., acomponent of additional model component(s) 1710 of IRM 1616).

FX conversion as implemented in IRM 1616 not only allows for theaggregation of risk for multi-currency portfolios and for the subsequentcollection of the initial margin in a single currency, it also accountsfor the market risk inherent in FX rates (albeit in a univariatemanner). Each currency pair is treated as a Risk Factor and as such isscaled in the same manner as any RF as described above.

The Portfolio P&L vector in Base currency is calculated by applying theappropriate scaled FX rate as follows:

$\begin{matrix}{\begin{bmatrix}{PnL_{\prod{;{\tau = W}}}} \\\ldots \\\ldots \\{PnL_{\prod{;{\tau = 1}}}}\end{bmatrix} = {\sum\limits_{i}{Q_{i} \cdot \begin{bmatrix}{F{X_{L;{\tau = W}}^{B} \cdot {Pn}}L_{i,{\tau = W}}} \\\ldots \\\ldots \\{F{X_{L;{\tau = 1}}^{B} \cdot {Pn}}L_{i,{\tau = 1}}}\end{bmatrix}}}} & (100)\end{matrix}$

where Q_(i) is the quantity (position) held of instrument i, FX_(L; τ)^(B) is the scaled FX rate (corresponding to scenario τ) that convertsinstrument P&L from its local currency L to Initial Margin Base currencyB. Using the pairwise product operator

., .

that represents pairwise multiplication across corresponding scenarios,the above equation can be rewritten more compactly as

$\begin{matrix}{{PnL_{\prod}} = {\sum\limits_{i}{Q_{i} \cdot \left\langle {{PnL}_{i},{FX}_{L}^{B}} \right\rangle}}} & (101)\end{matrix}$

here FX_(L) ^(B) represents a vector of FX rate scenarios {FX_(L; τ)^(B)}_(τ=1) ^(W). The Initial Margin IM_(Π) expressed in Base currencyis then obtained using equation (99) with P&L vector PnL_(Π). As statedfollowing equation (99), the Initial Margin may be capped at zero.

Currency-Based Risk Weights for Margin Allocation

For multicurrency portfolio there may also be an additional step ofallocating initial margin for each local currency group. This stepcalculates currency-based risk weights, which may be performed by thecurrency allocation component.

The local currency P&L vectors PnL_(i) may be used and may be separatedby their local currency groups. The Portfolio level P&L of all positionsin the same currency L=l is:

$\begin{matrix}{{PnL_{\Pi;l}} = {\sum_{i}{Q_{i} \cdot \left\langle {{{Pn}L_{i}},{F\; X_{L}^{B}}} \right\rangle \cdot 1_{L = l}}}} & (102)\end{matrix}$

where 1_(L=l) is the indicator function that takes on a value of onewhen the local currency is l and zero otherwise. As a check, thePortfolio level P&L should equal the sum of the P&L vectors separated bytheir original Local currencies:

$\begin{matrix}{{PnL_{\Pi}} = {\sum_{l}{PnL_{\Pi;l}}}} & (103)\end{matrix}$

The Initial Margin IM_(Π; l) for each local currency group l in the Basecurrency is obtained as the percentile at the level 1−q of its P&Lvector PnL_(Π; l).

The Initial Margin IM_(Π; l) for each currency group 1 may be computed,and the currency-based risk weights may be generated according to thesimplified formula:

$\begin{matrix}{{A_{l} = \frac{{IM}_{\Pi;l}}{\sum\limits_{l}^{\;}\;{IM}_{\Pi;l}}},\ {{\sum\limits_{l}A_{l}} = 1}} & (104)\end{matrix}$

The weights A_(l) are later used to calculate the final Initial Marginfor each Local currency group L=1.

Treatment of Correlation Risk by Correlation Stress Component (CSC)

The portfolio based Initial Margin described above assumes that thehistorical correlation over the VaR lookback window is expected to berelatively stable. This is somewhat justified by the choice of therelatively long lookback window. There exists, however, the risk thatthe realized correlations deteriorate over the holding period. This riskmay be accounted for by IRM 1616 with the estimation of the CorrelationStress Component (CSC) (e.g., a component of additional modelcomponent(s) 1710 of IRM 1616) described below.

The net effect on risk of any correlation changes may depend on thecomposition of the portfolio. A decrease in realized correlation tendsto decrease portfolio risk where positions are directional and tends toincrease portfolio risk where positions are primarily long/short,hedged. Thus a change to realized correlation could either make theportfolio more diversified (less risky), or it could make the portfolioless diversified (more risky). The empirically observed diversificationmay fail to materialize over the holding period, so that instead ofacting in a diversified manner the portfolio becomes riskier.

Directional portfolios where the P&L vectors of each position are highlycorrelated are expected to receive low diversification benefits, whichmakes the correlation risk insignificant in comparison to the volatilityrisk for such a portfolio. To illustrate using an extreme case, adirectional portfolio where the realized P&L correlation across all ofits individual positions is 100% can be expected to receive nodiversification benefits from IRM 1616. Such a portfolio would notrequire an additional method to account for correlation risk—any changeto realized correlation can only reduce the risk of the portfolio.

On the other hand, hedged portfolios carry a risk of becoming lesscorrelated over the holding period. The Correlation Stress Componenttakes such conditions into account, by estimating the portfolio IM underconditions of zero correlation between the individual instrumentpositions within the portfolio. Analogous with a variance calculation ofa sum of uncorrelated variables, the sum of the squares of the IM ofindividual instrument positions is determined, and the square-root ofthis sum is obtained. This provides the estimate for a totallyun-correlated scenario, denoted as IM_(CS):

$\begin{matrix}{{IM}_{CS} = {- \sqrt{\sum\limits_{i = 1}^{N}\;\left( {IM}_{\Pi;i} \right)^{2}}}} & (100)\end{matrix}$

Each IM_(Π; i) is calculated using equation (99) from the position PnLvector within the portfolio: PnL_(Π; i)=Q_(i)·

PnL_(i), FX_(L) ^(B)

. The Initial Margin with the Correlation Stress Component is thencalculated as follows:

$\begin{matrix}{{I\; M_{\Pi;{CS}}} = {{I\; M_{\Pi}} + {\min\left( {0,{w_{CS} \cdot \left( {{I\; M_{CS}} - {IM}_{\Pi}} \right)}} \right)}}} & (106)\end{matrix}$

where w_(CS) is the Correlation Stress Weight calibrated as discussedfurther below. Note that the same instrument P&L scenarios are used inthe calculation of IM_(Π) and IM_(CS).

As is clear from equation (106), the CSC will increase when thevolatility levels increase while the correlation among instruments doesnot change substantially. This is related to the fact that at increasedvolatility levels the cost and risk of de-correlation becomes higher.Moreover, the CSC does not react immediately to changes or breaks incorrelation; instead it provides a buffer against a potentialcorrelation break.

-   -   It should be noted that the CSC does not address the correlation        risk for directional portfolios where the P&L vectors of the        individual instruments are correlated only at moderate or low        levels. Such portfolios could become more correlated over the        holding period. However, typically the correlation risk for such        portfolios is dominated by the volatility risk that is accounted        for by IRM 1616.

Diversification Benefits Component

IRM 1616 further limits the amount of portfolio offsets formulti-product portfolios via a diversification benefits component (e.g.,a component of additional model component(s) 1710 of IRM 1616). Bylimiting the Diversification Benefits (DB), IRM 1616 helps mitigate therisk that the correlation to be realized in the future over the marginperiod turns out to be “worse” (riskier) than what is contained in thehistorical data. As recognition of diversification benefits is anadvantage of the margin model (IR 1616) that helps to allocate riskcapital more efficiently, the purpose is not to reduce DB excessively orto eliminate it.

In some examples, the DBs may be considered, according to regulatoryguidelines, as being that the amount of “margin reduction” should be nogreater than a predefined percentage (e.g., 80%) of the sum of theInitial Margins calculated with no offset. Moreover, it is recognized byregulatory guidelines that within certain predefined groups of financialinstruments, in some examples, 100% DB are permitted as products aresufficiently similar to provide a highly effective risk offset.

To limit the multi-product portfolio offsets to a predefined percentage(e.g., 80%,) the sum of the Initial Margins across all defined productgroup sub-portfolios is determined:

$\begin{matrix}{{IM}_{DB} = {\sum\limits_{g}{IM}_{\Pi;g}}} & (107)\end{matrix}$

where IM_(Π; g) is the Initial Margin for the product group gsub-portfolio, calculated using equation (99) with the position P&Lvector:

$\begin{matrix}{{PnL_{\Pi;g}} = {\sum\limits_{i}{Q_{i} \cdot \left\langle {{PnL}_{i},{FX}_{L}^{B}} \right\rangle \cdot 1_{G = g}}}} & (108)\end{matrix}$

where 1_(G=g) is the indicator function that takes on a value of 1 whenthe product group is g, and is zero otherwise.

The Initial Margin with the Correlation Stress Component and theDiversification Benefit Component is then the calculated as follows:

$\begin{matrix}{{IM}_{\Pi;{{CS} + {DB}}} = {\min\left( {{IM}_{{\Pi;{CS}}\;}{w_{DB} \cdot {IM}_{DB}}} \right)}} & (109)\end{matrix}$

where w_(DB) is the Diversification Benefit Weight set to a predefinedpercentage (e.g., 20%).

Final Initial Margin and Margin Allocation

The Initial Margin (IM) can now be rewritten as follows:

$\begin{matrix}\begin{matrix}{{IM}_{\Pi;{{CS} + {DB}}} = {\min\left( {{IM}_{\Pi;{CS}},{w_{DB}.\ {IM}_{DB}}} \right)}} \\{= {\min\left( {{{IM}_{\Pi} + {\min\left( {0,{w_{CS} \cdot \left( {{IM}_{CS} - {IM}_{\Pi}} \right)}} \right)}},{W_{DB}{IM}_{DB}}} \right)}} \\{= {{IM}_{\Pi} + {\min\left( {0,{w_{CS} \cdot \left( {{IM}_{CS} - {IM}_{\Pi}} \right)},{{w_{DB} \cdot {IM}_{DB}} - {IM}_{\Pi}}} \right)}}}\end{matrix} & (110)\end{matrix}$

The final Initial Margin may be generated either in the APC Off or APCOn configuration. In the APC Off (respectively, APC On) configuration,all margins that appear in equation (110) are calculated using APC Off(respectively, APC On) scenarios. The APC On configuration may beconfigured to comply with the applicable anti-procyclicality (APC)regulatory requirements. In some examples, the APC Off configuration maybe used for testing purposes.

The Initial Margin IM_(Π) computed using equation (99) may be capped atzero, and this ensures that the Initial Margin computed using equation(110) may also be capped at zero.

To allocate the Initial Margin for each currency group L=l, the InitialMargin IM_(Π; CS+DB; L=l) for each Local currency group L=l is allocatedusing the weights computed by equation (104):

$\begin{matrix}{{IM}_{\Pi;{{CS} + {DB}};{L = l}} = {A_{l} \cdot {IM}_{\Pi;{{CS} + {DB}}} \cdot {FX}_{B;T}^{L = l}}} & (111)\end{matrix}$

where FX_(B; T) ^(L=l) is the FX rate for converting from Base CurrencyB to Local on the Value Date T.

Generation of LRC Value(s) by LRC Model

Referring next to FIGS. 24-26, LRC model 1618 is described, with respectto steps 1904-1906 (FIG. 19). In particular, FIG. 24 is flowchartdiagram illustrating an example method of determining an LRC value forat least on portfolio; FIG. 25 is a functional block diagramillustrating a more detailed signal flow of the process for determiningthe LRC value by LRC model 1618; and FIG. 26 is an example illustrationof grouping methods for determining the LRC value, according to anon-limiting embodiment.

The LRC value is the sum of a Concentration Charge (CC) levied againstrelatively large cleared positions and a Bid-Ask Charge (BAC) applicableto all positions. The LRC itself is additive to the IRM 1616 and may beexpressed in the same currency as the IRM 1616 for single currencyportfolios, or a base currency for multi-currency portfolios.

As an overview, an example method of determining an LRC value by LRCmodel 1618 may include, performing the following method steps,illustrated in FIG. 24 and in more detail in FIG. 25. At step 2400, afirst synthetic portfolio of representative instruments may be createdbased on a delta technique, for example, by synthetic portfoliogenerator 1802. At step 2402, a second synthetic portfolio ofrepresentative instruments may be created based on a VaR technique, forexample, by synthetic portfolio generator 1802. Thus, by steps 2400 and2402, for any given portfolio, two portfolio representations may becreated using representative Instruments (RI) (with one based on theDelta method and the other based on the VaR method).

At step 2404, a concentration charge may be determined (for example byconcentration charge component 1804) for the concentrated representativeinstrument positions, for each of the first and second syntheticportfolios (created at respective steps 2400 and 2402).

At step 2406, a bid-ask charge may be determined (for example, bybid-ask charge component 1806) for the representative instrumentpositions, for each of the first and second synthetic portfolios(created at respective steps 2400 and 2402).

At step 2408, an LRC value may be determined, for example, by liquidityrisk charge component 1808. The LRC value may be determined as the sumof the concentration charge and the bid-ask charge.

In the event of a clearing member's default, a central counterparty willneed to eventually close out the defaulter's portfolio as part of thedefault management process. However, the CCP may not be able to do sowithout incurring additional liquidation losses if, for example, theportfolio contains concentrated positions. To mitigate this liquidationrisk, CCPs need to require members to post additional collateral in theform of an add-on to the Initial Margin.

For LRC model 1618, a zero-impact approach may be adopted fordetermining the add-on value. Specifically, the Liquidity Risk Chargemay be calculated under the assumption that positions are graduallyliquidated at a constant rate without impacting the market. However theexact liquidation strategy is not known ex ante. Under the zero-impactapproach, it is assumed that liquidation of the position happens over aperiod of sufficient length to ensure that liquidation costs arenegligible. In this approach, there is no price impact, but there isincreased market risk, as the liquidation period is extended relative tothe MPOR. Under the zero-impact approach, the following may beestimated:

-   -   i. the amount of daily position liquidation that will not cause        any substantial price impact;    -   ii. the additional market risk incurred by the extension of the        liquidation period. The theoretical foundations of the LRC model        are first-order price sensitivities (delta) and Value-at-Risk        (VaR), which are used to build risk-equivalent portfolios of        instruments called Representative Instruments. Portfolio        concentration and bid-ask spreads are evaluated at the        Representative Instrument level.

Grouping Methods

The mapping of instruments and the subsequent grouping of portfolios isa fundamental part of LRC model 1618. The modelling approach for LRC isrisk-based as it assesses liquidation cost at the portfolio group level,rather than at the level of each individual instrument. This makes therisk modelling approach of LRC model 1618 similar to that of IRM 1616.

LRC model 1618 uses a grouping method for the risk equivalence techniquedefined further below. FIG. 26 illustrates an example grouping methodaccording to an exemplary embodiment. As shown in FIG. 26, one or moreinstruments 2602 are grouped within LRC model 1618 first into one ormore expiry buckets 2604, then grouped into one or more liquiditybuckets (LB) 2606, and finally grouped into one or more concentrationcharge (CC)/bid-ask charge (BAC) groups 2608. Once instrument(s) 2602are grouped, LRC value(s) may be determined for one or more products2610 (e.g., ETD interest rate products). From most granular to least,the grouping hierarchy is as follows:

-   -   Instrument 2602: an exchange traded contract specified by its        underlying reference, expiry schedule, as well as strike price        and maturity in the case of an option.    -   Expiry Bucket 2604: a collection of instruments such that the        expiry of each underlying instrument maps to the same nearest        relative expiry.    -   Liquidity Bucket 2606: a collection of instruments from one or        more Expiry Buckets that can be represented by a single        instrument, referred to as the Representative Instrument, in the        sense of price risk equivalence.    -   Group (CC or BAC) 2608: a collection of instruments from one or        more Liquidity Buckets.

Specifically, consider Instrument i that is part of Clearing Memberportfolio Π on any given Value Date (VD). A Representative Instrument R(i) is any instrument that is deemed to appropriately representInstrument i. The risk equivalence approach is to construct a positionin R(i) that has the same amount of risk as the set of Instrumentpositions that are being mapped to it. Risk, in this context, isassessed in two ways for this purpose: Delta and VaR, each methodproducing its own equivalent portfolio.

Every Representative Instrument is associated with a unique LiquidityBucket defined as B_(r)={i|R(i)=r}. That is, B_(r) is the group of allinstruments that map to the same Representative Instrument. Everyinstrument maps to a unique R(i), and trivially every R(i) maps back toitself.

Portfolio Representation

As mentioned above, LRC model 1618 is based on two distinct assessments:the Delta approach and the VaR approach. The rationale for both is toapproximate the price risk, in the sense of Delta or VaR, between anygiven Liquidity Bucket sub-portfolio and its Representative Instrumentposition.

In particular, using portfolio representations allows aggregation of therisk associated with instruments with different levels of liquidity, bychoosing a representative instrument that provides for an appropriateprice risk equivalent representation and is actively traded.

Using both approaches provides for some degree of conservatism in thecalculation of the charge. In particular, using the Delta method helpsto account for extreme price movements of deep out-of-the-money optionswhereas using the VaR method helps to take into account volatility andcorrelation risk.

Portfolio Representation Delta Method

The purpose of the Delta method is to reproduce the first order (delta)price risk of any given portfolio using only Representative Instruments.

At a summary level, the steps in the Delta method are to calculate thefirst-order price sensitivity for each instrument position of theportfolio, aggregate the sensitivities across all instruments within therelevant liquidity bucket, then create a synthetic position in aRepresentative Instrument with the same aggregate delta sensitivity. Atthis stage the portfolio of the instruments in the liquidity bucket hasthe same delta as the Representative Instrument position. This isrepeated for all liquidity buckets.

First-order price sensitivity may be determined as follows:

-   -   1. If i is a futures derivative Instrument, its Instrument Delta        is calculated as

$\begin{matrix}{\Delta_{i} = {DV01_{i}}} & (112)\end{matrix}$

-   -    where DV01_(i) is the cash P&L impact of a one basis point        parallel shift of the underlying risk-factor curve, and where        the quantity held in i is a single lot.    -   2. If i is an option derivative Instrument, the Instrument Delta        is calculated as

$\begin{matrix}{\Delta_{i} = {DV0{1_{U{(i)}} \cdot {{sign}\left( \Delta_{\%} \right)} \cdot {\max\left( {{\Delta_{\%}},\Delta_{\%\mspace{14mu}\min}} \right)}}}} & (113)\end{matrix}$

-   -    where Δ_(%) is the Option Delta that represents the number of        units of the underlying U(i) that is held to remove the        first-order risk of one unit of the option instrument i        calculated using standard analytical formulas; and Δ_(% min) is        a floor on Δ_(%).

With price sensitivities Δ_(i) defined for each instrument in aLiquidity Bucket B_(r), the quantity Q_(r) ^(<D>) (number of lots) forthe Representative Instrument assigned to B_(r) may be determined sothat

$\begin{matrix}{{Q_{r}^{\langle D\rangle} \cdot \Delta_{r}} = {\sum\limits_{i \in B_{r}}{Q_{i} \cdot \Delta_{i}}}} & (114)\end{matrix}$

where Q_(i) is the quantity, positive for a long position and negativefor a short position, of Instrument i held in Liquidity Bucket B_(r).(Here, representative instruments may be selected to have the samedenominate currency as the instruments they represent.) Thequantity-weighted Instrument Delta of a Representative Instrument r isequated to the aggregated quantity-weighted delta of the group ofInstruments that map to the same R (i). At this, stage the positions ofthe individual instruments in the Liquidity Bucket are represented by asingle position in the Representative Instrument.

This technique is repeated for each liquidity bucket, so that asynthetic portfolio is created that is comprised entirely ofrepresentative instrument positions that each have the same delta as thegroup of instrument positions they represent.

Portfolio Representation-Value-at-Risk

The VaR method for constructing synthetic representative instrumentportfolios is next described. The purpose of the VaR method is torepresent the liquidity bucket sub-portfolios in a risk-based way thattakes into account volatility and correlation. Note that unlike in theDelta method, option positions are included directly (in the P&L vector)instead of being represented by their corresponding underlying futuredelta position.

The VaR method, denoted as Risk Measure (RM), is defined as the 1^(st)percentile of the P&L vector, or =PERCENTILE.INC(PnL_(i), 1−q) in thelanguage of Microsoft Excel®, where q is the VaR confidence level.

For each instrument i in a portfolio, the 1-day P&L vector PnL₁ iscomputed on the Value Date using the IRM 1616 Baseline parametersettings, as described above:

$\begin{matrix}{{PnL_{i}} = {\begin{bmatrix}{PnL_{i,{\tau = W_{o}}}} \\\ldots \\\ldots \\{PnL_{i,{\tau = 1}}}\end{bmatrix}.}} & (115)\end{matrix}$

where W₀ is the VaR lookback window.

The P&L vectors that belong to the same Liquidity Bucket are aggregatedas follows:

$\begin{matrix}{{PnL_{\Pi;B_{r}}} = {\sum\limits_{i \in B_{r}}{Pn{L_{i} \cdot Q_{i}}}}} & (116)\end{matrix}$

where Q_(i) is the quantity of instrument i held in portfolio Π andB_(r) is the Liquidity Bucket defined above. That is, for allInstruments assigned to the same Liquidity Bucket, the aggregate P&L isthe sum of the individual P&L vectors weighted by the instrumentquantities.

The quantity in the Representative Instrument is denoted by

and is calculated to have the equivalent RM as the Instrument positionsit represents. There are two possible solutions for this quantity: along position or a short position. The chosen solution is based onmatching the Risk Measure as well as portfolio directionality. Regardingdirectionality, a long or short position in the RepresentativeInstrument is selected so that its P&L vector is positively correlatedwith that of the liquidity bucket sub-portfolio.

The quantity in the Representative Instrument r is thus chosen so thatthe following two conditions are satisfied:

$\begin{matrix}{{{R{M\left( {PnL_{\Pi;B_{r}}} \right)}} = {R{M\left( {Q_{T}^{\langle V\rangle} \cdot {PnL}_{r}} \right)}}},{{{Correlation}\left( {{{Pn}L_{\Pi;B_{r}}},{Q_{r}^{\langle V\rangle} \cdot {PnL}_{r}}} \right)} \geq 0}} & (117)\end{matrix}$

The correlation for any two P&L vectors PnL_(i) and PnL_(j) is thePearson product-moment correlation and corresponds to the=CORREL(PnL_(i), PnL_(j)) function in Microsoft Excel®. Thesub-portfolio P&L for the liquidity bucket and the P&L of theRepresentative Instrument held in quantity

are required to be equivalent in terms of the Risk Measure anddirectionality.

The quantity

in the Representative Instrument satisfying conditions (117) is given by

$\begin{matrix}{Q_{r}^{\langle V\rangle} = \left\{ \begin{matrix}{\frac{R{M\left( {PnL_{\Pi;B_{r}}} \right)}}{R{M\left( {PnL_{r}} \right)}},} & {{Corr_{r}} \geq 0} \\{­\ \frac{R{M\left( {PnL_{\Pi;B_{r}}} \right)}}{R{M\left( {{- P}nL_{r}} \right)}}\ ,} & {otherwise}\end{matrix} \right.} & (118)\end{matrix}$

where Corr_(r)=Correlation(PnL_(Π; B) _(r) , PnL_(r)) is the correlationbetween the Representative Instrument P&L (1 unit long) and theLiquidity Bucket portfolio P&L.

Once Representative Instrument quantities have been estimated for eachliquidity bucket under each method, these quantities are used in the CCand BAC determinations.

Concentration Charge

The Concentration Charge is calculated twice for each group: once usingthe Delta method portfolio representation, and then again using the VaRmethod portfolio representation—both as described herein.

It is assumed for each portfolio that price risk, ignoring concentrationrisk, is covered within the MPOR used in IRM 1616. Any RepresentativeInstrument position that is expected to take longer than the MPOR to beliquidated, referred to as Excess Quantity

, is calculated as

$\begin{matrix}{Q_{r}^{\langle E\rangle} = {{{sign}\left( Q_{r} \right)} \cdot {\max\left( {\left| Q_{r} \middle| {{- h} \cdot {CT}_{r}} \right.,0} \right)}}} & (119)\end{matrix}$

where h is the holding period as per IRM 1616, CT_(r) is theConcentration Threshold (CT) for instrument r (for simplicity, in thenotation of Q_(r), the Portfolio Representation method used (Delta orVaR) is suppressed). The Concentration Threshold CT_(r) represents thequantity that can be traded in one day without any price impact. Thecalibration of CT is described below.

To determine the period of time for the liquidation of Excess Quantity

, the Liquidation Period (LP) for an individual RepresentativeInstrument excess position is obtained as the Excess Quantity divided bythe CT:

$\begin{matrix}{{LP_{r}} = {\min\left( {\frac{Q_{t}^{\langle E\rangle}}{CT_{r}}\ ,{\max\left( {0,{{timeToExpiry}_{r} - h}} \right)}} \right)}} & (120)\end{matrix}$

where LP is expressed in units of days and where CT, in some examples,is floored at a strictly positive constant. The Liquidation Period iscapped by the time-to-expiry (defined as the number of days between thevalue date and the expiry date) of the Representative Instrument, exceptfor Representative Instruments with physical delivery, which are notcapped since such instruments continue to have risk after expiry.

The Concentration Charge is assessed at the Concentration Charge Group(CCG) level. For each CCG a synthetic portfolio of Excess Quantities inRepresentative Instruments is constructed. Portfolio offsets are allowedbetween positions within the CCG.

The Concentration Charge at the CCG level is calculated under theassumption that the portfolio positions are gradually liquidated at adaily rate equal to the level of Concentration Threshold CT_(r).Analogous with a variance calculation of a sum of uncorrelatedvariables, the Concentration Charge is calculated as the square root ofthe sum of squared Risk Measures of the remaining portfolio on each dayin the maximum Liquidation Period:

$\begin{matrix}{{CC}_{CCG_{k}} = \sqrt{\sum\limits_{t = 0}^{{LP_{\max}^{{CCG}_{k}}} - 1}\left( {{RM}\left\lbrack {\sum\limits_{r \in {CCG_{k}}}\left( {Q_{r}^{\langle{E,t}\rangle} \cdot {PnL}_{r}} \right)} \right\rbrack} \right)^{2}}} & (121)\end{matrix}$

where

represents the remaining quantity of the Representative Instrument r onday t of the maximum Liquidation Period:

$\begin{matrix}{Q_{r}^{\langle{E,t}\rangle} = {sig{{n\ \left( Q_{r}^{\langle E\rangle} \right)} \cdot {\max\left( {0,\ {{Q_{t}^{\langle E\rangle}} - {t \cdot {CT}_{r}}}} \right)}}}} & (122)\end{matrix}$

LP_(max) ^(CCG) ^(k) denotes the maximum Liquidation Period across theRepresentative Instruments in CCG_(k) rounded up to the nearest integer:

$\begin{matrix}{{LP_{\max}^{CCG_{k}}} = \left\lceil {\max_{r \in {CCG_{k}}}\left( {LP_{r}} \right)} \right\rceil} & (123)\end{matrix}$

where ┌⋅┐ is the ceiling operator rounding the value up to the nearestinteger.

A special treatment in equation (121) may be applied to cash-settledRepresentative Instruments that expire within the corresponding maximumLiquidation Period for some index t=t_(exp) (corresponding to the expiryday) such that t_(exp)<LP_(max) ^(CCG) ^(k) . Since such instruments donot represent any risk after expiry, they may be excluded from thecalculation in equation (121) for t>t_(exp) by setting

=0 for t>t_(exp). In some examples, this treatment may not apply toRepresentative Instruments with physical delivery (e.g., bond futures).

The Concentration Charge for a given portfolio is calculated as

$\begin{matrix}{{{CC_{\Pi}} = {\sum\limits_{k}{\max\left( {CC_{CCG_{k\prime}}^{\langle D\rangle}CC_{CCG_{k}}^{\langle V\rangle}} \right)}}}.} & (124)\end{matrix}$

That is, the Concentration Charge is the sum across all ConcentrationCharge Groups, where each of the charges is represented by the maximumof two monetary values calculated using the VaR and the Delta method.

Note that the choice of the maximum of the two CC values calculatedusing the Delta and VaR method is one of conservatism. Additionalconservatism is also achieved by the fact that offsets betweenRepresentative Instrument positions are not allowed across the differentCCGs.

Bid-Ask Charge

The BAC component captures a different aspect of liquidity risk that isnot addressed by CC. This is why BAC is fully additive to anyConcentration Charges. It is possible at any given time for the CCP toface portfolio liquidation costs regardless of the level of instrumentposition concentration. This motivates BAC as a separate and distinctcharge.

As with the CC component, BAC is computed using two separate riskassessments—one based on the Delta and one based on the VaRrepresentations of the portfolios.

For a given portfolio Π, the Bid-Ask Charge is calculated for eachBid-Ask Charge Group (BACG), once

and

are calculated for each Representative Instrument. In contrast with theCC calculation, BAC is not based on the excess quantity beyond athreshold, but rather the entire position (in the RepresentativeInstrument) qualifies for BAC because the risk associated with thebid-ask spread is not taken into account in the IRM 1616 model.

The BAC calculation separately treats long and short positions in theRepresentative Instruments. The first argument inside the max operatorapplies to long portfolios, the second to short:

$\begin{matrix}{{BAC_{BACG_{k}}^{\langle m\rangle}} = {{\max\left\lbrack {{\sum\limits_{\underset{Q_{r}^{\langle m\rangle} > 0}{{r \in {BACG}_{k}},}}\left( {Q_{r}^{\langle m\rangle} \cdot {FX}_{L;r}^{B} \cdot \frac{BA_{r}}{2}} \right)},{\sum\limits_{\underset{Q_{r}^{\langle m\rangle} < 0}{{r \in {BACG}_{k}},}}\left( {{- Q_{r}^{\langle m\rangle}} \cdot {FX}_{L;r}^{B} \cdot \frac{BA_{r}}{2}} \right)}} \right\rbrack}.}} & (121)\end{matrix}$

where m=D or V, depending on whether the Delta or VaR method is used forportfolio representation,

is the quantity, FX_(L; r) ^(B) is spot exchange rate from the localcurrency of Representative Instrument r to the Base Currency B, andBA_(r) is the Bid-Ask spread parameter for Representative Instrument r.

The division by two in BA_(r)/2 is used to get from the mid-price toeither bid or ask price. Taking the maximum in equation (125), of longonly and short only positions, is appropriate as positions may be tradedas a strategy as opposed to individual positions.

The BAC assessed on portfolio Π is the sum of the individual pairwisemaxima of the BAC Group charges:

$\begin{matrix}{{{BAC_{\Pi}} = {\sum\limits_{k}{\max\left( {{BAC_{BACG_{k}}^{\langle D\rangle}},\ {BAC_{BACG_{k}}^{\langle V\rangle}}} \right)}}}.} & (126)\end{matrix}$

Note again that the choice of the maximum of the two BAC valuescalculated using the Delta and VaR method is one of conservatism.Additional conservatism is also ensured because offsets betweenRepresentative Instrument positions are not allowed across differentBACGs.

Total Liquidity Risk Charge

The total Liquidity Risk Charge assessed on portfolio H is the sum ofthe Concentration Charge and the Bid-Ask Charge:

$\begin{matrix}{{LRC_{\Pi}} = {{CC_{\Pi}} + {BA{C_{\Pi}.}}}} & (127)\end{matrix}$

Synthetic Data Generation by Synthetic Data Generator

Next, the generation of synthetic datasets is discussed. In someexamples, the synthetic datasets may be generated by synthetic datagenerator 1626.

The use of synthetic data in testing an initial margin model has severalpractical advantages including: providing data sets that can be designedto emulate historical data; providing multiple sets of out-of-sampledata for testing; and that the data sets themselves may be technicallyunlimited (e.g., an arbitrarily large sample size).

Synthetic data provides the flexibility to create customized data setsthat explore model behavior under conditions that may be controlled byan analyst. This can take the form of creating data sets that match thehistorical data record on one or more desired metrics, for examplevolatility or other given moments of a distribution includingcross-moments; or creating data that exhibit the characteristics ofpre-defined events such as stress scenarios. The model may thencalibrated to the synthetic data and evaluated using backtesting todetermine whether model performance continues to be acceptable.

The Benign Normal dataset is well behaved with respect to volatility(constant), autocorrelation (none), correlation (constant), and returndistributions (normal). It represents a baseline dataset used as part ofthe backtesting framework that IRM 1616 is expected to pass. The RegimeChange dataset aims to simulate the characteristics that may be observedin the historical data, for example, volatility clustering,autocorrelation in returns, heavy tails in the distributions, butcontrols at any time specifically the regime where the stress event isisolated to start, for example, at the 2001^(st) observation and end atthe 2500^(th) observation. Both synthetic data sets may be designed fortesting at the risk factor level, therefore certain features of the datasuch as restrictions on the levels or movements of the yield curve maynot be preserved.

Both linear and non-linear (option) risk factors may be generated usingthe same basic methodology. Referring next to FIG. 27, a flowchartdiagram illustrating an example method of generating synthetic data isdescribed. At step 2700, inter-dependence between all risk factors ismodeled with a copula function. This is also referred to as specifyingthe joint distribution or multivariate distribution. Step 2700 includessub-steps 2702-2708, described further below).

The second step (step 2710), includes specifying the marginaldistributions or the distribution of the univariate risk factorsthemselves. This can be performed either by specifying a parametricdistribution or by using a volatility model. Step 2710 includessub-steps 2712-2714, described further below).

Note that the process of generating the option risk factors is subjectto the following constraints: meaningful correlation (positive definite,as well as term structure) and volatility (term structure) of underlyingfuture RFs and volatility RFs; reasonable shape for the volatility smileat each expiry; and non-arbitrage relation in time along the ATM strikes

To achieve the first constraint, the correlation and volatilitystructure from historical risk factor data is used in step 2700. Thesecond constraint and third constraint are achieved by enforcingarbitrage free conditions on the volatility smile at each expiry whichis done in step two (step 2710). The second step (step 2710) may beperformed such that the final dataset can be tailored for needs of aparticular analysis. For example, a dataset can be constructed to testhow an initial margin model handles abrupt changes in volatility.

At step 2716, one or more multi-regime datasets may be created. At step2718, post-processing of the generated dataset(s) may be performed toinverse transform a set of risk factors to an original format. At step2720, the post-processed dataset(s) may be stored as syntheticdataset(s), for example, in database(s) 1624.

Specification of the Joint Distribution

For the specification of the joint distribution (as part of step 2700),the following high level steps are performed. At sub-step 2702,transformation of historical option risk factors to a predefined set ofrisk factors may be performed. Sub-step 2702 may include transformingoption risk factors into their forward volatility, risk reversal, andbutterfly representations. Step 2700 may also include creating a copulato model joint distribution, including defining a historical period forfitting, calculating a dependence measure and selecting and generatingthe copula.

Transformation of historical option risk factors (sub-step 2702) may beused to maintain a reasonable shape for the smile curve at each tenorand also to capture a non-arbitrage relation in time along the ATMstrikes. At sub-step 2702, the option risk factors may be transformed toa new (predefined) set of risk factors including: forward vol v_(E; t),risk reversal r_(E; t) and butterfly b_(E; t). The forward vol for thefirst tenor (shortest tenor) is the ATMF risk factor at the same tenor:

$\begin{matrix}{v_{E_{1};t} = \sigma_{0,{E_{1};t}}} & (128)\end{matrix}$

The forward volatility risk factor for the subsequent tenors is thefollowing:

$\begin{matrix}{{v_{E_{i};t} = \sqrt{\frac{{E_{i} \cdot \left( \sigma_{0,{E_{i};t}} \right)^{2}} - {E_{i - 1} \cdot \left( \sigma_{0,{E_{i - 1};t}} \right)^{2}}}{E_{i} - E_{i - 1}}}},{i = 2},3,4} & (129)\end{matrix}$

The risk reversal risk factor is calculated for each tenor E_(i) asfollows:

$\begin{matrix}{{r_{E_{i};t} = {\sigma_{1,{E_{i};t}} - \sigma_{{- 1},{E_{i};t}}}},{i = \ 1},2,3,4} & (130)\end{matrix}$

The butterfly risk factor is calculated for each tenor E_(i) as follows:

$\begin{matrix}{{b_{E_{i};t} = {\sigma_{1,{E_{i};t}} - {2 \cdot \sigma_{0,{E_{i};t}}} + \sigma_{{- 1},{E_{i};t}}}},{i = \ 1},2,3,4} & (131)\end{matrix}$

Thus after the transformation, instead of option risk factors at 3strikes for each tenor, a forward volatility, risk reversal, andbutterfly risk factor at each tenor are generated.

Regarding copulas, in general, an n-dimensional copula can be describedas a multivariate distribution function defined on the unit cube[0,1]^(n), with uniformly distributed marginal. In an exampleembodiment, a Gaussian copula is selected for the Benign Normal datasetand a t-copula is selected with the number of degrees of freedom set to5 (e.g., to ensure a reasonably strong tail dependence) for the RegimeChange dataset.

The following equations describe the Gaussian copula and t-copula:

$\begin{matrix}{{C_{R}^{Ga}(u)} = {\Phi_{R}^{n}\left( {{\Phi_{1}^{- 1}\left( u_{1} \right)}\ ,\ldots\mspace{14mu},{\Phi_{n}^{- 1}\left( u_{n} \right)}} \right)}} & (132) \\{{C_{R,v}^{t}(u)} = {\Psi_{R,v}^{n}\left( {{\Psi_{v}^{- 1}\left( u_{1} \right)},\ldots\mspace{14mu},\ {\Psi_{v}^{- 1}\left( u_{n} \right)}} \right)}} & (133)\end{matrix}$

The dependence measure chosen in constructing the copulas is linearcorrelation (Pearson correlation). Given a historical time period forfitting, the Pearson's correlation is computed for the matrix R. Themethod by which a linear correlation matrix R is applied to the jointdistribution to enforce dependence is the Cholesky decomposition. Thistechnique is performed for both the Benign Normal and Regime Changedatasets.

To create the copula representing the joint distribution:

-   -   1. Transform option risk factors into their forward vol, risk        reversal, and butterfly representations (sub-step 2702).    -   2. Filter the data appropriately given the input date range for        historical fitting period (sub-step 2704).    -   3. Calculate absolute returns for the risk factors (sub-step        2706). Mean center and scale the returns to variance=1.    -   4. Calculate the dependence measure, the Pearson's correlation        matrix R (e.g., for the historical return data).    -   5. Generate copulas for N synthetic draws (sub-step 2708)—either        Gaussian copulas or t-copulas:        -   a. For each draw, generate a random multivariate            distribution. For a Gaussian copula, this is generated from            the multivariate normal distribution. For a t-copula, this            is generated from the multivariate t distribution.        -   b. Apply the linear correlation R to the multivariate            distribution with Cholesky decomposition.        -   c. Transform the multivariate distribution to have uniform            marginal distributions using the appropriate pdf.

Specification of Marginal Distributions

For the specification of the marginal distributions (step 2710) thefollowing high level steps may be performed:

-   -   1. Fit the historical data for the fitted parametric marginals        and the volatility model marginals (sub-step 2712).    -   2. Use the copula to generate the marginal distributions        incrementally (sub-step 2714)        -   a. Calculate synthetic return for a single day        -   b. For option risk factors            -   i. Apply mean reverting algorithm (only for the Regime                Change dataset)            -   ii. Check for constraints and no-arbitrage conditions.                If a violation exists, discard return and simulate                another set of returns. If no violation exists, move to                simulate the next date.

For an example methodology, an unconditional mean model is chosen wherethe mean is set to zero. The GJR-GARCH(1,1) model is also chosen for thevolatility model. Thus the equations for generating the Regime Changesynthetic data are:

$\begin{matrix}{X_{t} = {{\mu_{t} + {ɛ_{t}\mspace{14mu}{where}\mspace{14mu} ɛ_{t}}} = {\sigma_{t}Z_{t}}}} & (134) \\{\mu_{t} = 0} & (135) \\{\sigma_{t}^{2} = {\omega + {\alpha_{1}ɛ_{t - 1}^{2}} + {\gamma_{1}I_{t - 1}ɛ_{t - 1}^{2}} + {\beta_{1}\sigma_{t - 1}^{2}}}} & (136)\end{matrix}$

In the fitting process for this model, the following parameters arefound by MLE for each time series: ω, α₁, β₁, γ₁. The shape and skewparameters to specify the NIG innovations are also found for each timeseries in the fitting process.

Mean Reverting Component and No Arbitrage Conditions for Option RiskFactors

To maintain meaningful correlation of volatility of underlying futureRFs and volatility RFs, the dynamics of the simulated forward vol riskfactor {tilde over (v)}_(E; t) may be modeled as a time-dependent Wienerprocess with a mean-reverting component. In contrast, the risk reversal{tilde over (r)}_(E; t) and butterfly {tilde over (b)}_(E; t) riskfactors may be simulated directly from the fitted parametricdistribution or the volatility model return output without themean-reverting component. For the forward vol risk factor, first thesimulated return d{tilde over (v)}_(E; t) are determined at the currenttime step. The mean of the historical forward vol risk factor v _(E) isthen used to calculate the mean reverting component:

$\begin{matrix}{{{\overset{˜}{v}}_{E_{i};t} = {{\overset{˜}{v}}_{E_{i};{t - 1}} + {d{\overset{˜}{v}}_{E_{i};t}} + {\alpha_{MR} \cdot \left( {{\overset{¯}{v}}_{E_{i}} - {\overset{˜}{v}}_{E_{i};{t - 1}}} \right)}}},\ {i = \ 1},2,3,4} & (137)\end{matrix}$

Here α_(MR) is a mean reverting parameter set, for example, to 0.05. Forsimulation with no mean-reverting of the forward vol risk factor, setα_(MR)=0. Once the simulated forward vol, risk reversal, and butterflyrisk factors are computed, the constraint and no-arbitrage checks areperformed. These checks include the following:

$\begin{matrix}{{\overset{˜}{v}}_{E_{i};t} > {minvol}} & (138) \\{{\overset{˜}{b}}_{E_{i};t} > 0} & (139) \\{{\overset{˜}{\sigma}}_{{S = {- 1}},{E_{i};t}} > {minvol}} & (140) \\{{\overset{˜}{\sigma}}_{{S = 1},{E_{i};t}} > {minvol}} & (141)\end{matrix}$

The minimum volatility value minvol is set to 0.

Applying Marginal Distributions to Generate Synthetic Data Incrementally

To generate the synthetic data incrementally (sub-step 2714), thefollowing process may be performed:

1. For the fitted parametric marginals:

-   -   a. Fit each historical risk factor return series to the chosen        parametric distribution using maximum likelihood estimation        (MLE). This would result in the fitted parameters for either a        normal distribution (location and scale parameters) or the NIG        distribution (location, scale, shape, and skew parameters).

2. For the volatility model marginals:

-   -   a. Fit each historical risk factor return series to the        GARCH(1,1) model. This would result in the GARCH parameters for        the volatility model and the parameters for the innovations        (shape and skew parameters for NIG innovations).

3. Using the previously generated copula, generate M synthetic returnsincrementally.

-   -   a. Select an initial risk factor level for all risk factors.    -   b. Using the copula uniform marginals generate a set of returns        for a single day        -   i. For the fitted parametric marginals:            -   1. Apply the inverse CDF of the parametric distribution                given the fitted parameters found in 1(a) to simulate                the risk factor return. In the case of generating the                benign normal dataset, the mean is set to zero and the                MLE fitted standard deviation is used. (Alternatively,                for the benign normal dataset, the simulated returns can                be directly calculated by multiplying the correlated                multivariate normal by either the MLE fitted standard                deviation or a sample standard deviation for each risk                factor.) In addition, the transformed option volatility                risk factor returns may be multiplied by a constant                (e.g., 0.01). (For the benign normal dataset, the                volatility of the option risk factors may be reduced so                that the constant volatility of the underlying is                reflected, i.e., the volatility of the implied                volatility is relatively small)        -   ii. For the volatility model marginals:            -   1. Create the corresponding innovations from the uniform                copula marginals using the innovation parameters found                in 2(a) by applying the inverse CDF and then simulate                the risk factor return using the GARCH(1,1) model and                the innovation.    -   c. Apply mean reverting algorithm for option risk forward vol        risk factors.        -   i. Compute historic forward vol return mean.        -   ii. Adjust simulated forward vol returns with mean-reverting            component.    -   d. Check for option risk factor constraints. If a violation        occurs, discard the returns generated in 3(b). If no violation        occurs then keep these returns, calculate the current risk        factor level by adding the return to the previous date's risk        factor level, and move to simulating the next date.    -   e. Repeat steps 3(b) to 3(d) until M returns that satisfy the        given constraints are generated. Note that since some of the        copula innovations are discarded because of violations, more        than M innovations (e.g. M+1250) are required to create M        returns.

Inverse Transformation Back to Volatility Risk Factors

For post processing (step 2718), the option risk factors may betransformed back to their original format.

Once the forward volatility, risk reversal, and butterfly option riskfactors are simulated, an inverse transformation is applied to obtainsynthetic option risk factors in their original format:

$\begin{matrix}{{\overset{˜}{\sigma}}_{0,{E_{1};t}} = {\overset{˜}{v}}_{E_{1};t}} & (142) \\{{{\overset{˜}{\sigma}}_{0,{E_{i};t}} = \sqrt{\frac{{E_{i - 1} \cdot \left( {\overset{˜}{\sigma}}_{0,{E_{i - 1};t}} \right)^{2}} + {\left( {E_{i} - E_{i - 1}} \right) \cdot \left( {\overset{˜}{v}}_{E_{i};t} \right)^{2}}}{E_{i}}}},{i = 2},3,4} & (143) \\{{{\overset{˜}{\sigma}}_{{- 1},{E_{i};t}} = \frac{\left( {{\overset{˜}{b}}_{E_{i};t} + {2 \cdot {\overset{˜}{\sigma}}_{0,{E_{i};t}}} - {\overset{˜}{r}}_{E_{i};t}} \right)}{2}},{i = 1},2,3,4} & (144) \\{{{\overset{˜}{\sigma}}_{1,{E_{i};t}} = \frac{\left( {{\overset{˜}{b}}_{E_{i};t} + {2 \cdot {\overset{˜}{\sigma}}_{0,{E_{i};t}}} + {\overset{˜}{r}}_{E_{i};t}} \right)}{2}},{i = 1},2,3,4} & (145)\end{matrix}$

Model Calibration Module

Next, the calibration of model parameters for IRM 1616 is discussed. Insome examples, the calibration of the model parameters may be performedby model calibration module 1628. Wo model parameters, the EWMA Lambdaparameter and the Correlation Stress Weight, are discussed in detailbelow. However, it is understood that calibration of any other suitableparameters of IRM 1616 may be performed by model calibration module1628. As discussed above, calibration may be performed by a user (e.g.,an analyst, an administrator, etc.) in communication with modelcalibration module 1628 via user interface 1634.

In some examples, model calibration module 1628 may IRM 1616 calibratethe Lambda (λ) parameter of IRM 1616 associated with the EWMA process onan in-sample basis. Lambda may be used to adapt IRM 1616, for example,to current market volatility dynamics as per the EWMA process. IRM 1616

Once Lambda is calibrated, model calibration module 1628 may calibratethe Correlation Stress Weight (w_(CS)) parameter of IRM 1616. Thus, thecalibration of Lambda and w_(CS) may be performed, in some examples,sequentially rather than jointly. This may be appropriate as Lambda isdesigned to process univariate Risk Factors, while w_(CS) targets thedependence structure between Risk Factors. Additionally, the EWMAvolatility estimates σ_(RV; t) for a given value date may berecalculated after calibration.

On any given value date in production, a user (e.g., a risk analyst)using IRM 1616 can take the calibrated values into account as one factorin determining the desired values of Lambda and Correlation StressWeight. The user, via user interface 1634, can either accept or rejectthe calibrated values to update the parameters for IRM 1616, forexample, based on current market conditions and any impact on clearingmember portfolios.

The lambda parameter (λ) may be calibrated to meet a set of specifiedobjectives within sample, and may be evaluated for performance on thosesame metrics out of sample. In an example configuration, P&L Rounding isset to off, the Volatility Floor is set to on, the Stress VolatilityComponent is set to on, the APC is set to off, the Range of λ is set to0.965 to 0.995, the Calibration increment is set to 0.005, the In-SamplePeriod is set to 500 days, the Calibration Frequency is set to 60 daysand the Holding Period is set to 1 day. Once the calibration iscompleted, the calibrated Lambda is used for a predefined time period(e.g., the next 60 days). At the end of the time period (e.g., 60 days)another 500 day in-sample calibration may be completed with thecalibration period ending on the current value date. In this manner, theModel may perform in-sample calibration at intervals corresponding tothe calibration frequency. In this example, the portfolios used for theA calibration are the outright linear Risk Factor portfolios, with atotal of 688 portfolios of this type (344 long and 344 short).

The calibration may be subject to the following two risk managementconstraints being satisfied:

-   -   Basel Traffic Light (BTL) Reds: that the number of BTL Red        portfolios does not exceed the threshold PA^((Red)) given by        equation (155).    -   Basel Traffic Light Yellows: that the number of BTL Yellow        portfolios does not exceed the threshold PA^((Yellow)) given by        equation (156).        The calibration objective may be formulated as follows:    -   Average Coverage Ratio Minimization: that IRM 1616 seek a value        of λ that minimizes the average Coverage Ratio (described        further below) across the calibration portfolios.

One purpose of the calibration is to find the set of parameters, firstLambda then the Correlation Stress Weight, that meet predefined riskmanagement constraints—that is, to ensure with a high degree ofcertainty (e.g., at least 99%) that the initial margin exceeds inmagnitude the realized variation margin over a margin period of risk. Inan example embodiment, in the calibration procedure, the primary riskmanagement concerns become the constraints, while the secondaryobjective related to the Coverage Ratio becomes the objective function.

Constraint Relaxation Process

In cases where the constraints identified above cannot be satisfied overthe admissible range of parameters, the calibration process may beadjusted as follows:

Relaxation #1:

-   -   Objective function: minimize the number of BTL Yellows above the        threshold PA^((Yellow)). In case of a tie, i.e., if the number        of BTL Yellows is the same for several Lambda values, choose the        Lambda value that yields the smallest Average Coverage Ratio.    -   Constraint: Number of BTL Reds does not exceed the threshold        PA^((Red)).

If the constraint in Relaxation #1 can be satisfied, a calibrated valueof Lambda may be obtained. Otherwise, the following may be considered:

Relaxation #2:

-   -   Objective function: minimize the number of BTL Reds above the        threshold PA^((Red)). In case of a tie, i.e., if this number is        the same for several Lambda values, choose the value for which        the number of the BTL Yellows above the threshold PA^((Yellow))        is smallest. If there is still a tie, i.e., if the number of the        BTL Yellows is the same for several Lambda values, choose the        Lambda value that yields the smallest Average Coverage Ratio.    -   No Constraints

Since Relaxation #2 does not contain any constraints, it is guaranteedthat a calibrated value of Lambda may be obtained. Empirically it hasbeen observed that relaxations occur infrequently.

In one example, in calibrating Lambda based on historical data,relaxations were not been triggered in any of a 37 calibration periods.

A process similar to the Lambda calibration may be followed for thecalibration of the Correlation Stress Weight w_(CS). The differencesbetween calibration of Lambda and the Correlation Stress Weight are inthe calibration configuration and the calibration portfolios. In anexample configuration, P&L Rounding is set to off, the Volatility Flooris set to on, the Stress Volatility Component is set to on, the APC isset to off, the Range of w_(CS) is set to 1% to 5%, the Calibrationincrement is set to 0.5%, the In-Sample Period is set to 500 days, theCalibration Frequency is set to 60 days and the Holding Period is set to1 day. In this example, the portfolios used for the Correlation StressWeight w_(CS) calibration include Risk Factor portfolios representingadjacent spread and butterfly strategies (both long and short). The useof spread and butterfly strategies may help to generate values of w_(CS)that appropriately take account of non-directional risk portfolios. Inthis example, the Risk Factors include STIR, Bond Yield, Repo, OIS Swap,and Vanilla Swap Risk Factors.

Generally, the error of the VaR (percentile) estimator decreases as thenumber of available observations increases, which supports a longer,multi-year period lookback period. Moreover, for shorter VaR windows(e.g., 500 days), the initial margin is likely to exhibit unstablebehavior. A longer lookback window further allows IRM 1616 to capture agreater range of market conditions (including periods of stress).

The slope of the margin curve above is treated as proxy for marginprocyclicality. If the margin level is insensitive to the market stresslevel (a slope of zero), the margin may be considered to benon-procyclical. The desired margin behavior may be configured to meetthe following constraints:

-   -   As market stress levels are increasing, the total amount of        margin should not decrease. That is, downstream systems such as        a Clearing House systems should not be actively returning margin        as market conditions are deteriorating.    -   When the market stress level is above the level corresponding to        the APC Index percentile (i.e. when the APC buffer begins to be        eroded), the procyclicality of the final IM should be no worse        than that of Base IM.

Model Testing Module

Model testing module 1630 may be configured to perform one or more testsover one or more testing categories for IRM 1616. In a non-limitingexample embodiment, model testing module 1630 may be configured toperform one or more tests from among seven (7) testing categories. Theseven testing categories may include:

-   -   1) Fundamentals Testing: Testing to explore the fundamental        characteristics of the data and IRM 1616 components.    -   2) Backtesting: Comparing the IRM 1616 output (the IM) with        actual market outcomes (calculated VM). The Backtesting process        may be configured to assess model performance using        out-of-sample data (e.g. data not used during the calibration        process).    -   3) Procyclicality Testing: Backtesting using metrics designed to        assess how well IRM 1616 addresses procyclicality.    -   4) Sensitivity Testing:        -   a. Rolling Backtest Analysis: Assesses a performance of IRM            1616 over changing market conditions including periods of            stress via backtesting with a rolling window.        -   b. Parameter Sensitivity Analysis: Assesses changes in the            output of IRM 1616 due to a change in an input parameter.    -   5) Incremental Testing: Starting from a most basic model, the        impact on the output of IRM 1616 may be evaluated from the        incremental addition of model components.    -   6) Model Comparison with Historical Simulation: Comparison of        the output of IRM 1616 with the output of the historical        simulation model.    -   7) Assumptions Backtesting: Backtesting related to specific        Model assumptions.

Table 2 indicates how testing categories are paired with portfolios, aswell as data that may be used for the various testing. Additionaldescription of the different portfolios tested is provided below.

TABLE 2 Testing Overview of IRM 1616 Testing Category Portfolios DataFundamentals RF Historical Backtesting RF, Synthetic Instruments,Historical, Real Instruments Synthetic Procyclicality RF, SyntheticInstruments, Historical, Real Instruments Synthetic Sensitivity RealInstruments Historical Incremental Real Instruments HistoricalComparison with Historical Real Instrument Historical SimulationAssumptions Real Instrument Historical

Synthetic datasets are created from random draws from a specifieddistribution. This is described in detail above. In the example, thedatasets are created to mimic historical Risk Factors and are designedfor specific testing purposes. There are two types of synthetic datasets used in testing:

-   -   Benign Normal Datasets (Backtesting only)—The benign synthetic        data has the same number of Risk Factors as the historical Risk        Factor data. The data is generated from a multi-variate normal        distribution with empirically observed constant volatilities and        a constant correlation. This data is designed to test the        Initial Margin Model with no shocks coming from the data.    -   Regime Change Datasets (Backtesting only)—The regime change        synthetic data also has the same number of Risk Factors as the        historical Risk Factor data. This data is generated using a        t-copula for the joint distribution and a GARCH model for the        marginal distributions. The data contains a regime of volatility        stress based on the historical time period 2007-2009 surrounded        by regimes of relative calm. The dataset is designed to test the        Initial Margin Model with a scenario of changing volatility and        changing correlation.

The construction of testing portfolios follows a “building-block”approach. This approach aims to test particular parts of the modelworkflow to ensure proper functionality as well as to expose IRM 1616 toreal world portfolios. The portfolios range from simple constructions tomore complex: the most basic portfolios are Risk Factor “outright”portfolios, which test how IRM 1616 handles single risk factors with nocomplexities related to pricing. One set of the more complex portfoliosare the clearing member portfolios which help ensure proper initialmargin coverage at an aggregated level.

Portfolio groups may include synthetic risk factor, synthetic(non-linear) instrument and real instrument groups. With synthetic riskfactor groups, Risk Factor portfolios may contain outright Risk Factorsor linear combinations thereof. In the example, pricing functions arenot used to generate P&L distribution with these portfolios. The VM onday t is calculated by subtracting the Risk Factor value on day t fromthe Risk Factor value corresponding to the day at the end of the holdingperiod.

With synthetic instrument groups, stylized theoretical optioninstruments may be created to mimic a constant time to maturity anddelta risk profile. The portfolio holdings may be rolled back on eachday to their specified time to maturities (e.g., 3M to expiry) and totheir specified relative strike (e.g., ATM+1). For VM calculations theportfolio holdings may be fixed over the holding period, i.e. theabsolute strikes are kept unchanged. The VM on day t is calculated bysubtracting the portfolio value on day t from the portfolio valuecorresponding to the day at the end of the holding period. In someexamples, IRM 1616 may perform a full revaluation, so that each optionfor each scenario may be fully repriced.

With real instrument groups, portfolios may contain exchange listedcontracts (e.g., “real instrument positions”) rolling through time buthaving consistent exposure (e.g. W1, W2, etc.). Pricing functions may beused to generate P&L distribution. Position holdings may be rolled tothe next expiry 2 days ahead of the respective nearest contract expiry(e.g. STIR futures may roll monthly or quarterly, repo futures may rollmonthly). Absolute option strikes may be adjusted daily as specified bytheir relative strikes (e.g. ATM+1). For VM calculations, the portfolioholdings may be fixed over the holding period (for example, absolutestrikes may be kept unchanged for option instruments). The VM on day tis calculated by subtracting the portfolio value on day t from theportfolio value corresponding to the day at the end of the holdingperiod.

In some examples, IRM 1616 may be evaluated against three types ofportfolios: one or more stylized strategies, one or more randomportfolios and one or more real clearing member(s) (CM) portfolios.

For stylized strategies, theoretical portfolios may be composed by riskfactors, synthetic and/or real instruments, which may follow a specifictrading strategy throughout their existence so as to be comparable overtime. In some examples, some of the items of the theoretical portfolios(e.g., risk factors, synthetic instruments, real instruments) may alsobe designed based on the positions of one or more CMs.

For random portfolios, hypothetical portfolios may be configured thatcould conceivably be exposed to, but that might not be represented by,actual observed portfolios or specific stylized strategies. Theportfolio weights in random portfolios may be generated randomlyaccording (for example) to a uniform distribution withweight_(i)˜U[−1,1], and with positive (negative) values indicating along (short) position. Once generated, the weights in a random portfoliomay remain fixed to allow for meaningful comparison over time. Theportfolio components include synthetic Risk Factors and/or instruments.

For real CM portfolios, positions as observed historically in theaccounts of one or more clearing members may be used. This set ofportfolios may be useful to assess the performance of IRM 1616 as if ithad been in place at the clearing house system during the testingperiod.

The available days for backtesting a portfolio may vary depending on theportfolio type. In general, portfolio types may be tested against theentire history available. For certain portfolios or product types, theperiod for backtesting may be reduced because of the availability ofdata related to portfolio construction and/or data related to VM. Forexample, bond futures may have less data available for VM than otherproducts and the backtest period may be much shorter for other products.

The following symbols are used to denote different backtest periods ofthe historical data for a non-limiting example of testing of IRM 1616:

-   -   B₀ as the (full) available backtest period (Sep. 3, 2007 to Feb.        14, 2018)    -   B_(CM) as the (full) available backtest period for Clearing        Member backtests (May 1, 2013 to Feb. 14, 2018)    -   B_(S) as the backtest period for a chosen stress period (Sep. 3,        2007 Aug. 10, 2009)

Fundamentals Testing Category

Fundamentals testing may be used to analyze key properties of the RiskFactor time series data on which IRM 1616 is based, and may provideevidence that modeling choices are consistent with the data. Propertiesof the Risk Factor time series data may include, without being limitedto, autocorrelation, volatility clustering, heavy tails, weak stationaryand zero versus non-zero mean. For example, (linear) autocorrelations ofreturn time series are often insignificant. Regarding volatilityclustering, time series of squared returns (and magnitudes of returns)typically show significant autocorrelation. Regarding heavy tails,return series are typically leptokurtic (or heavy tailed). Regardingweak stationarity, returns are typically (weakly) stationary, which mayallow for meaningful forecasting. Regarding zero vs. non-zero mean, theconditional mean of returns is often statistically insignificant.Accordingly the one-day (unsealed) RF return time series may be analyzedin light of the above characteristics. In the example, log returns areused for option RFs and FX RFs whereas absolute returns are used for alllinear RFs.

The RF returns are assessed to determine whether the returns exhibitsignificant autocorrelation based on the 1250 day period from Apr. 17,2013 to Feb. 16, 2018. The Ljung-Box test with 10 lags is used to testthe null hypothesis that all autocorrelations up to lag 10 are zero.

The RF returns are assessed to determine they the returns exhibitvolatility clustering, which is manifested as the positiveautocorrelation in the squared returns based on the 1250 day period fromApr. 17, 2013 to Feb. 16, 2018. The significance of theseautocorrelations at the 5% level of significance is tested using theLjung-Box test with 10 lags.

The one-day RF returns are assessed to determine whether the returnsexhibit heavy tails by examining the (unconditional) excess kurtosiscomputed based on the period from Sep. 25, 2002 to Feb. 16, 2018.Together with the presence of autocorrelation in squared returns, theheavy tails are indicative of volatility clustering, which suggests thata filtering approach is appropriate (i.e., as opposed to the simplerHistorical Simulation approach).

The RF returns are assessed to determine whether the returns are weaklystationary, i.e., if they maintain a constant unconditional mean,variance and autocorrelation, computed based on the period from Sep. 25,2002 to Feb. 16, 2018. The presence of weak stationarity is importantfor meaningful volatility forecasting.

Unit root tests are used including the Phillips-Perron and augmentedDickey-Fuller (ADF) test with 10 lags. The null hypothesis for both unitroot tests is that the time series has a unit root, so that a smallp-value supports stationarity. For each of the 476 RFs, the p-values forboth of the unit root tests applied are significant at the 5%significance level, supporting weak stationarity.

The RF returns are assessed to determine whether the returns exhibitstatistically significant means using a rolling analysis over the periodfrom Aug. 3, 2007 to Feb. 16, 2018 with a window of 1250 days. TheStudent's t-test is applied to the (unsealed) RF returns at the 5%significance level. The null hypothesis for the test is that the truemean return is zero; hence a small p-value would indicate a significantnon-zero mean.

The presence of volatility clustering and heavy tails of the RF returnssuggests that a filtering approach may be desired to explain theobserved volatility behavior and produce an accurate volatilityforecast. The EWMA filtering approach may be compared with the moregeneral GARCH(1,1) filtering approach by performing the residualsdiagnostics for the fitted models and by applying the Akaike InformationCriterion.

In the case of EWMA, the model equations are used:

$\begin{matrix}{{r_{t} = {\mu + {\sigma_{t}ɛ_{t}}}},\ {\sigma_{t}^{2} = {{\lambda\sigma_{t - 1}^{2}} + {\left( {1 - \lambda} \right)r_{t - 1}^{2}}}}} & (146)\end{matrix}$

whereas in the case of GARCH the model equations are used:

$\begin{matrix}{{r_{t} = {\mu + {\sigma_{t}ɛ_{t}}}},\ {\sigma_{t}^{2} = {\omega + {\beta\sigma_{t - 1}^{2}} + {\alpha r_{t - 1}^{2}}}}} & (147)\end{matrix}$

where the error term ε_(t) is assumed to be normally distributed withmean 0 and variance 1 in both models.

The EWMA model and GARCH model are fitted to each of the RF return timeseries using the Maximum Likelihood Estimation (MLE) principle, so thateach RF has its own set of fitted model parameters. The 1250 day periodfrom Apr. 17, 2013 to Feb. 16, 2018 is used. For both EWMA and GARCH,the model parameter μ is fitted. For EWMA an additional parameter λ isfitted and for GARCH(1,1), the three additional parameters α, β, ω arefitted.

To test the assumption of independence of the residuals ε_(t), theLjung-Box test is used with 10 lags (for the absence of autocorrelation)at the 5% significance level. The test is applied to the residuals andsquared residuals, respectively. Both models help to reducesubstantially the number of significant autocorrelations in the squaredresiduals when compared to the no filtering model

The Kolmogorov-Smirnov test at the 5% significance level is used toassess if the residuals follow the distribution specified in the model(e.g., normal distribution). The null hypothesis for theKolmogorov-Smirnov test is that the two distributions underconsideration are the same.

The two models are further compared using the Akaike InformationCriterion (AIC), which evaluates the model's goodness-of-fit adjustedfor the number of model parameters. The AIC formula is used withnormalization by the sample size N:

${{AIC} = {{- \frac{\log\left( {likelihood} \right)}{N}} + \frac{2k}{N}}},$

where k is the number of model parameters. The AIC criterion impliesthat a more complex model with a larger number of parameters isjustified if the difference in optimized log likelihoods is bigger thanthe number of additional parameters.

In comparing the EWMA with GARCH(1,1) models, both EWMA and GARCH(1,1)generally help to explain the RF behavior. It is observed thatGARCH(1,1) performs better in some aspects of model evaluation (e.g.,autocorrelation in squared residuals) while EWMA has a comparableperformance in some other aspects (e.g., goodness-of-fit, AIC). Thusoverall EWMA appears to be an adequate parsimonious model. EWMA may bemore desirable from the point of view of operational simplicity, as theparameters of the GARCH(1,1) model would likely need to be adjusted onthe individual RF basis (in contrast with EWMA where the same calibratedLambda can be applied for all RFs) and may need more frequentrecalibration.

Next, the use of a single Lambda across all RFs in EWMA filtering isdemonstrated as being appropriate and that the basic EWMA model with asingle Lambda (without any additional components) performs reasonablywell at the univariate RF level. Specifically, the basic EWMA model(described above) is considered (e.g., without any such additional modelcomponents 1710 as volatility floor, stress volatility, etc.) with λfitted for each Risk Factor using the Maximum Likelihood Estimationprinciple. In this example, the error term is assumed to have Student'st-distribution with the number of degrees of freedom set equal to 3, andλ is re-fitted on each backtest day. This model is compared with thebasic EWMA model based on a single Lambda set equal to 0.98 across allRFs an all backtest days.

The volatility forecast {circumflex over (σ)}_(T) is capped at themaximum historical volatility σ_(max; T). It is examined how often thevolatility cap kicks in by being the smaller of the two arguments of theminimum function, i.e., how often it is observed that

$\begin{matrix}{\sigma_{\max;T}\  < {\max\left( {{\sigma_{{RV};T} + {\Delta\sigma_{{stress};T}}},\sigma_{{F{loor}};T}} \right)}} & (148)\end{matrix}$

in the APC Off configuration, and

$\begin{matrix}{\sigma_{\max;T} < {\max\left( {{\sigma_{{RV};T} + {\alpha_{{APC};T} \cdot {\max\left( {{\Delta\;\sigma_{{RV};T}},{{APC}_{buffer} \cdot \sigma_{{RV};T}}} \right)}}},\sigma_{{Floor};T}} \right)}} & (149)\end{matrix}$

in the APC On configuration. The study is performed for the historicaldata set across all RFs in the APC On and APC Off configurations, overthe period from Sep. 3, 2007 to Feb. 14, 2018.

It is determined that the volatility cap does not kick in for any of theRFs in the APC Off configuration.

The treatment with de-meaning and re-meaning vs. no mean treatment (theadopted treatment in IRM 1616) are compared for the purpose of RFscenario generation. Since one of the objectives of de-meaning is toremove the drift/mean from the returns, a t-test for zero mean isapplied to a set of the RF scenario returns computed with de-meaning anda corresponding set of the RF scenario returns computed without any meantreatment. The null hypothesis for the test is that the mean is zero;hence a small p-value would indicate a significant non-zero mean.

The EWMA formula for the variance term uses an initialization, or “seed”value, for the first variance term at time τ=1 to start the recursionprocess. The initial return at that same time also appears in theequation but does not need an initialization as the first available datapoint in FD is already defined in terms of returns.

Different seed values could conceivably result in different volatilityestimates, leading to different Initial Margin estimates as the scenariodistributions are affected by volatility estimates. The choice of theseed value and whether the seed value may cause a statisticallysignificant difference (at the standard 5% significance level) in thedistribution of the Risk Factor scenarios are examined.

The hypothesis that a different seeding method does not produce astatistically different distribution of the Risk Factor scenarios istested. As a reasonable alternative seeding method, the seed value isset equal to the square of the initial return within the lookbackwindow. The Kolmogorov-Smirnov test is applied to compare thedistribution of the RF scenario returns computed with the adopted EWMAseeding method and the alternative method. The null hypothesis for theKolmogorov-Smirnov test is that the two distributions underconsideration are the same. The analysis is performed over the periodfrom Sep. 3, 2007 to Feb. 14, 2018, with W₀=1250 scenarios on each day.

The adopted square-root-of-time time scaling rule (discussed above)) maybe justified in the absence of autocorrelations in the risk factorreturns. As observed above, some of the RFs exhibit significantautocorrelations. However, the performance of the chosen time scalingmethod is still expected to be adequate since for the majority of theRFs the magnitude of autocorrelation coefficients is small or onlymoderately large.

Accordingly, a 2-day Risk Factor backtesting of the basic EWMA model(with a fixed Lambda of 0.98) is performed where the time scaling isperformed using the square-root-of-time rule.

Next, the behavior of the RF APC indices during high volatility and lowvolatility conditions is examined.

To further assess the adequacy of the RF APC Indices, the procyclicalityprofiles of the basic EWMA vol σ_(RV; T) vs. EWMA vol are compared withthe APC buffer σ_(T) defined as:

$\begin{matrix}{{\hat{\sigma}}_{T} = {\sigma_{{RV};T} + {\alpha_{{APC};T} \cdot {APC}_{buffer} \cdot \sigma_{{RV};T}}}} & ({l50})\end{matrix}$

As a measure of procyclicality an analog of the APC Expected Shortfallis used (with the initial margins being replaced with volatilities inequation (168)). It may be observed that the APC buffer indeed helps toimprove the procyclicality profile as manifested in the reduced APCExpected Shortfall for all Risk Factors.

Correlation analysis is conducted to assess the reliability andsignificance of correlations between different interest rate instrumentsover time. An instrument is understood here as a set of contractsgrouped by product category and currency. The statistical properties ofproducts corresponding to the same interest rate instrument areconsidered as indicative of reliable and significant correlations,including resilience during stressed historical periods. The reliabilityof the correlation between two instruments may be measured by the changein the correlation between the corresponding representative productsover the holding period. Two instruments are significantly correlated ifthe correlation between the corresponding representative products isstatistically significant (from zero).

Pairs of representative products may selected that are part of anydesired types of products for use with IRM 1616. In this example, twosets of pairs are considered: one set where the two instruments belongto the same ESMA Margin Group, and the other set with pairs where thetwo instruments belong to different ESMA Margin Groups.

The correlation analysis is conducted for 1-day instrument returns on arolling (overlapping) window basis over the historical period from Sep.3, 2007 to Feb. 14, 2018, inclusive of available stress events. Thelength of the rolling window is set to 1250 days in line with the choiceof the lookback window W₀ in IRM 1616. Settlement data for instrumentreturns is used when available; otherwise instrument returns derivedfrom the Risk Factors is used. Both Pearson correlation levels andchanges in Pearson correlation over the 2-day holding period areexamined.

Next, the impact of the scaling process on correlations among the RiskFactor returns is examined. In particular, this scaling process includesthe Volatility Floor and Stress Volatility components of IRM 1616. Forthe analysis, Risk Factor pairs corresponding to spreads in the set ofStylized Synthetic (Linear) Risk Factor portfolios are considered,limiting the analysis to the spreads that contain exactly two RFs. Inthis example, the total number of such spreads is 247.

For each RF pair, Pearson and Kendall tau correlations are determinedbefore and after for each business date from Sep. 3, 2007 to Feb. 14,2018. Whereas the Pearson correlation measures the degree of lineardependency, the Kendall tau correlation measures the ordinal association(i.e., rank correlation) between the two quantities. The scaledcorrelations for each date are computed based on the 1,250 RF returnscenarios scaling (APC On, 1-day holding period) on that date; theunscaled correlations are evaluated based on the 1-day RF returns in alookback window of 1,250 days for the corresponding date.

Next the proportion of static arbitrage instances (including call/putspread arbitrage, butterfly arbitrage, and calendar arbitrage) in theoption RF scenarios (APC On, 2-day holding period) is determined acrossall dates from Sep. 3, 2007 to Feb. 14, 2018 and all 129 option RiskFactors. This proportion is compared to the corresponding proportion ofarbitrage instances in the settlement data used to generate the RFs.

Next, what types of returns (absolute or logarithmic) are appropriatefor each group of the RFs (linear RFs, option RFs, and FX) aredetermined. Specifically, it is examined for what return type the scaledRF returns exhibit better stationarity, homoscedasticity (constantvariance) and have less autocorrelation. This study is non-parametricand is thus consistent with the fact IRM 1616 does not impose anyparametric constraints of the distribution of the RF returns.

Two sets of 1250 RF return scenarios are generated on the backtest date(Feb. 14, 2018): one set is obtained using absolute returns for all RFsand the other is obtained using log returns for all RFs. As some linearRFs may be negative, in which case log returns may be undefined, onlythose linear RFs for which a full set of scaled log returns on Feb. 14,2018 are available are examined.

To assess weak stationarity, the Phillips-Perron test is used. The nullhypothesis for the test is that the time series has a unit root, so thata small p-value supports weak stationarity. To assess the significanceof autocorrelations, the Ljung-Box test with 10 Lags is applied to thescaled RF returns. To assess homoscedasticity, the Ljung-Box test with10 lags is applied to the squared RF return scenarios. In the presenceof homoscedasticity, it is expected to observe zero autocorrelation inthe squared RF return scenarios, in which case a small p-value is notobserved.

Backtesting Category

The backtesting framework compares the Initial Margin (determined by IRM1616) calculated for portfolios to the realized variation margin fordata that is out-of-sample, i.e. not used in the calibration of IRM1616. The expected coverage of IRM 1616 is related to the VaR confidencelevel. As per applicable regulatory requirements for ETD products, theconfidence level may be set to 99%, which implies that the VM can beexpected to exceed the IM (in magnitude) 1% of the time. The expectedexceedance probability is denoted as p=1−0.99. When the VM is strictlysmaller than the IM this is termed synonymously as an exceedance, or abreach. In order to evaluate if the observed performance of IRM 1616 isaligned with expectation, all portfolios (in some examples) arebacktested with several metrics calculated.

Let D denote the total number of backtest days and let d∈{1, . . . , D}be the backtest day index. The backtest days form the backtest period[FBD,LD]. The variation margin VM_(d) on backtest day d is defined asthe portfolio P&L over the margin period of risk, i.e., the differencebetween the portfolio value at the end of the MPOR and the portfoliovalue on day d.

Let {IM_(d)}_(d=1) ^(D) and {VM_(d)}_(d=1) ^(D) be the series of InitialMargins and variation margins, respectively, calculated for a portfolioover the backtest period. The breach indicator variable {I_(d)}_(d=1)^(D), is defined as

$\begin{matrix}{I_{d} = \left\{ \begin{matrix}1 & {if} & {{IM}_{d} > {VM_{d}}} \\0 & {if} & {{IM}_{d} \leq {VM_{d}}}\end{matrix} \right.} & (151)\end{matrix}$

for d∈{1, 2, . . . , D}. It is noted that IM≤0 so that a breach occurswhere IM is larger than VM. Let n₁ be the number of exceedances so thatn₁=Σ_(d=1) ^(D)I_(d) and let n₀=D−n₁ be the number of non-exceedances.

The Breach Size on backtest date d is defined as follows:

$\begin{matrix}{{{BS}_{d} = \frac{VM_{d}}{{IM}_{d}}},{{{if}\mspace{14mu} I_{d}} = 1}} & (152)\end{matrix}$

The metric is defined only for I_(d)=1, i.e. only on day d when a breachoccurs. If IM_(d) is zero then Breach Size is undefined.

To construct the time series of breach events, two time-seriesaggregation methods are used: the overlapping method and thenon-overlapping method. Let h∈{1,2} denote the holding period expressedin days, and let IM_(d) and VM_(d) denote the initial and variationmargin correspondingly on backtest date d, where d∈{1, 2, . . . , D}.The two aggregation methods are then defined as follows.

If the overlapping data method is applied, the pairs (IM_(d), VM_(d)) ofh-day IM and VM for successive backtest dates will be used to determinewhether a breach event occurs. Although such method does not reduce theoriginal sample size, it creates dependency in the data. This method isdefined only if h>1.

The non-overlapping method is used for h≥1. Aggregate the originaloverlapping VM and IM time series into h groups of non-overlapping timeseries, beginning from the earliest date in the backtest period. If D isa multiple of h then the size of each group is given by

$\begin{matrix}{N = \frac{D}{h}} & (153)\end{matrix}$

If D is not a multiple of h then let r be the remainder after dividing Dby h. Then the size of the groups 1 through r is

$\left\lfloor \frac{D}{h} \right\rfloor + 1$

and the size of the remaining groups is

$\left\lfloor \frac{D}{h} \right\rfloor$

(here └⋅┘ denotes the floor operator that rounds the number down to thenearest integer). For example, when h=2 and D=7 (where d includes datesd=1, 2, 3, 4, 5, 6 and 7), four (4) first groups (d=1, 3, 5 and 7)alternating with three (3) second (remaining) groups (d=2, 4 and 6) arecreated. The non-overlapping method does lead to the reduction of theeffective sample size; however, in contrast with the overlapping method,it does not create dependency in the data and thus standard statisticaltests can be applied to evaluate the performance.

Based on the portfolio composition for the IM calculation and the chosensource of VM, three types of backtesting may be performed: Risk Factor,settlement VM (SM VM) and risk factor derived VM (RF VM). The RiskFactor type of backtesting may include Risk Factor Portfolio P&L/IMand/or Risk Factor VM (RF Portfolios). In the Risk Factor type ofbacktesting, the VM is calculated directly from Risk Factor levelsrather than from actual instrument prices. In some examples, P&Lrounding may not be applicable to the Risk Factor portfolios since priceticks are not defined. SM VM and RF VM are both a type of Instrumentbacktesting. The Instrument backtesting may include Instrument PortfolioP&L/IM. For the Settlement VM type (e.g., many Real InstrumentPortfolios), the VM is calculated directly from instrument prices. Forthe Risk Factor Derived VM (e.g., Synthetic Instrument and select RealInstrument Portfolios), the SM VM may be used for portfolios whereSettlement VM may not exist or may not be tested. For consistency of thetest, P&L rounding may not be applied.

The following two large-scale configurations are defined when executinga complete backtest run (in the example below):

-   -   APC On or APC Off—Backtest with APC component turned on versus        turned off.    -   1Day or 2Day—Backtest with holding period set to 1 day versus        being set to a regulatory mandated 2 day ETD holding period.        Using these two configuration choices, the following four        combinations of backtest runs can be performed for all        portfolios: 1 Day-APC Off, 1 Day-APC On, 2 Day-APC Off, 2        Day-APC On, with the exception of Risk Factor Portfolios and        Non-Linear Synthetic Portfolios for which only 1Day-APC Off,        1Day-APC On runs are performed.

For the backtesting metrics, two types of metrics may be distinguished:Primary and Investigative. Primary metrics are assigned thresholds on aRed-Amber-Green (R-A-G) evaluation scale. Investigative metrics areadditional metrics that may further explain backtesting results.

For primary metrics, a Red-Amber-Green (R-A-G) evaluation scale isprovided. A “Green” outcome indicates the backtesting result does notsuggest an issue with the quality of the IRM 1616. An “Amber” outcomesuggests that while the IRM 1616 has an adequate performance overall, aninvestigation may be performed for that portfolio; while a “Red” outcomeindicates that there may exist a potential model issue and additionalanalysis may be performed. It is noted that portfolios where the totalcount of “Amber” is below the PA metric (discussed below; whereapplicable) are considered to perform within the accepted statisticaltolerance.

One of the primary metrics to evaluate IRM 1616 backtest results is theBasel Traffic Light (BTL) metric. The Basel Committee on BankingSupervision developed an evaluation framework for the probability ofobserving a particular number breaches. The rationale behind using thistest is to assess whether the unconditional coverage of the initialmargin model does not exceed the target breach probability.

The breach event on a backtest date can be modeled as a Bernoulli randomvariable. Hence, the cumulative number of breaches can be modelled as abinomial random variable. Given a confidence level 1−p (or equivalently,given the breach probability p), the probability of observing a numberof breaches not exceeding n₁ over D backtest days is given by

$\begin{matrix}{{F\left( {{n_{1}\text{;}D},p} \right)} = {\sum\limits_{i = 0}^{n_{1}}{\begin{pmatrix}D \\i\end{pmatrix}{p^{i}\left( {1 - p} \right)}^{D - i}}}} & (154)\end{matrix}$

where F is the binomial cumulative distribution function with parametersD, p. Three zones defined in the BTL test are Green, Yellow, and Red,with each zone corresponding to a range of the cumulative probabilityfor the binomial distribution:

-   -   Green: if F(n₁; D, p)<95%, i.e., n₁ corresponds to a cumulative        probability below 95%, then the portfolio passes;    -   Yellow: if 95%≤F(n₁; D, p)<99.99%, i.e., n₁ corresponds to a        cumulative probability at or above 95% but below 99.99%, then        the portfolio may require further investigation;    -   Red: if F(n₁; D, p)≥99.99%, i.e., n₁ corresponds to a cumulative        probability at or above 99.99%, then the portfolio fails the        test.

The BTL test is used in combination with either the non-overlappingmethod or the overlapping aggregation method. The application of the BTLtest may be straightforward when the overlapping aggregation method isused. For the non-overlapping aggregation method, the BTL test proceduredescribed above is applied to each of the h groups of non-overlappingtime series. The outcome is then determined based on the average valueof the cumulative probability across the groups.

Table 3 shows BTL zones expressed in terms of the cumulative probabilitythresholds as well as in terms of the number n₁ of breaches for a500-day sample (assuming overlapping aggregation method) at the 99% VaRconfidence level. If the true coverage probability of the model is 1%,it is expected that for a 500 day sample, there will be less than 9breaches for 95% of the portfolios tested (Green), assuming theindependence of all outcomes.

TABLE 3 BTL Outcomes on the R-A-G Scale Cumulative Probability Number ofBreaches R-A-G BTL Zone Thresholds 500 day sample, 99% VaR G Green <95% n₁ < 9  A Yellow >= 95% and <99.99% 9 ≤ n₁ < 15 R Red >= 99.99% n₁ ≥ 15

Since the power of the test is low in small sample sizes, the followingrules apply for different sample sizes:

-   -   D≥500—the test can be applied both in the case of 1-day and        2-day holding period    -   250≤D<500—the test can be applied in the case of 1-day holding        period, or in the case of 2-day holding period provided the        overlapping aggregation method is used    -   D<250—the test will not be conducted        The BTL test is one-tailed and hence may not be operational to        detect models with unconditional coverage that is too        conservative.

The BTL test is used in combination with either non-overlapping oroverlapping aggregation method. Each method may have some limitations.The use of the non-overlapping aggregation method may result in thereduction of the effective sample size when h >1. At the same time thebreach series can be assumed to be independent with this approach. Onthe other hand, the overlapping aggregation method may create dependencein the data although it does not reduce the original sample size. In thecase of 2-day holding period, the BTL results are reported based on thenon-overlapping aggregation method unless stated otherwise.

Next, a method to aggregate BTL results of a group of portfolios (e.g.,RI-Real CM, SI-Random, etc.) and determine whether the observed numbersof Yellow and Red portfolios are acceptable is described.

For a group of portfolios, let n*>1 be the total number of portfoliosand let N^((Yellow)) and N^((Red)) be the observed number of Yellow andRed portfolios, respectively. By assuming independence of BTL resultsacross different portfolios, the theoretical thresholds for the numberof Yellow and Red portfolios may be derived at the confidence levelp′=99.99%. Specifically, the thresholds “Yellow Portfolios Allowed”(PA^((Yellow))) and “Red Portfolios Allowed” (PA^((Red)) are defined as:

$\begin{matrix}{{PA^{({Red})}} = \left\lfloor {F^{- 1}\left( {{p^{\prime};\ n^{*}},{0.0001}} \right)} \right\rfloor} & (155) \\{{PA^{({Yellow})}} = {\left\lfloor {F^{- 1}\left( {{p^{\prime};\ n^{*}},0.05} \right)} \right\rfloor - \left\lfloor {F^{- 1}\left( {{p^{\prime};\ n^{*}},{0.0001}} \right)} \right\rfloor}} & (156)\end{matrix}$

where F⁻¹ is the inverse binomial cumulative distribution function and└⋅┘ denotes the floor operator that rounds the number down to thenearest integer. In one implementation of the inverse cumulativedistribution function of the binomial distribution, this function is inline with the Microsoft Excel® 2013 function binom.inv( ).

The observed numbers of Yellow and Red portfolios are considered asacceptable (at p′=99.99%.) if the following inequalities are satisfied:

$\begin{matrix}{N^{({Yellow})} \leq {PA^{({Yellow})}}} & (157) \\{N^{({Red})} \leq {PA^{({Red})}}} & (158)\end{matrix}$

A possible limitation of this approach is that it does not take intoaccount the existence of possible relations in the portfolio group.

Investigative metrics may be used when additional analysis ofbacktesting results is desired. The R-A-G thresholds are assigned whereappropriate to identify any outliers for further investigation.Portfolios where the total count of either Red or Amber is below therelevant PA metric are considered to perform within the acceptedstatistical tolerance.

The Kupiec test for unconditional coverage (also referred to as theproportion of failures test, or POF test) evaluates whether the observednumber of breaches is consistent with the desired coverage probabilityp. While serving a similar purpose to the BTL Test, Kupiec's POF Testcan provide an additional assessment of whether the model IM is tooconservative.

The null and alternative hypothesis to be tested is as follows:

$\begin{matrix}\left\{ \begin{matrix}{{H_{0}\text{:}{E\left\lbrack I_{d} \right\rbrack}} = p} \\{{H_{1}\text{:}{E\left\lbrack I_{d} \right\rbrack}} \neq p}\end{matrix} \right. & (159)\end{matrix}$

Under the null hypothesis the coverage is correct when the expectationof I_(b) equals the coverage probability. The test statistic is given by

$\begin{matrix}{{LR_{uc}} = {{{- 2}\;\log\;\left( \frac{L(p)}{L\left( \overset{\hat{}}{\pi} \right)} \right)} = {{- 2}\;\log\;\left( \frac{\left( {1 - p} \right)^{n_{0}}p^{n_{1}}}{\left( {1 - \overset{\hat{}}{\pi}} \right)^{n_{0}}{\overset{\hat{}}{\pi}}^{n_{1}}} \right)}}} & (160)\end{matrix}$

where

$\overset{\hat{}}{\pi} = \frac{n_{1}}{n_{0} + n_{1}}$

is an estimate of p. The test statistic LR_(uc) is asymptoticallydistributed as the chi-square distribution χ²(1) with 1 degree offreedom as the number of backtest days D→→. Note that this test istwo-tailed and that a failure can occur if the number n₁ of breaches istoo high (fails aggressively implying E[I_(d)]>p) or too low (failsconservatively implying E[I_(d)]<p). A failure occurs when the teststatistic exceeds the χ²(1) threshold value for a chosen significancelevel.

The p-value associated with the calculated test statistic LR^(uc)(p,{circumflex over (π)}) is found as follows:

$\begin{matrix}{{p\_{value}}_{LR_{uc}} = {P{r\left( {{LR_{uc}} \geq {L{R_{uc}\left( {p,\overset{\hat{}}{\pi}} \right)}}} \right)}}} & (161)\end{matrix}$

The test is performed at the 5% significance level and the followingthree possible outcomes are considered:

-   -   Pass if the p-value computed using equation (161) is greater        than or equal to 5%,    -   Fail Aggressive if the p-value is less than 5% and the observed        breach proportion {circumflex over (π)} is above p=1%, and    -   Fail Conservative if the p-value is less than 5% and the        observed breach proportion {circumflex over (π)} is below p=1%.

The Fail Conservative outcome is not regarded as an indication of aninappropriate model; instead, it suggests the model behavior isconservative as the realized breach probability is below 1%.Accordingly, the R-A-G scale shown in Table 4 may be preferred forreporting the Kupiec test results.

TABLE 4 R-A-G Threshold for Kupiec Test R-A-G Threshold G p-value ≥ 5%or {circumflex over (π)} < 1% A 0.01% ≤ p-value < 5% and observed breachproportion {circumflex over (π)} > 1% R p-value < 0.01% and {circumflexover (π)} > 1%

The Kupiec test is used in combination with either non-overlapping oroverlapping aggregation method. In case when the non-overlappingaggregation method is used, the test procedure described above isapplied separately to each of the h groups of non-overlapping timeseries, and the outcome is determined based on the average p-value takenacross the groups. The Kupiec test results for multiple portfolios canbe aggregated using a method analogous to theoverlapping/non-overlapping aggregation methods described above.

Possible limitations of the Kupiec test may include that the Kupiec testhas limited power for small sample sizes and may fail to detect VaRmeasures that systematically under-report risk. Therefore, in someexamples, the test may not be conducted if the sample size is less than250 observations. Moreover, the chi-square approximation used in thetest may be inaccurate for small coverage probabilities (equivalently,high confidence levels) and only moderately large sample sizes. As aresult, the actual significance level of the test, in some examples, maybe higher than the nominal (5%) significance level.

The Christoffersen test of independence is a likelihood ratio testaiming to assess if breach events are independent of each other. If not,the breaches can cluster (especially in periods of stress).

The Christoffersen test for independence tests whether the sequence ofbreaches is independent, i.e. that the occurrence of a breach on day dis independent of the occurrence of a breach that occurs on day d−1. Thenull and alternative hypothesis to be tested are thus specified asfollows:

$\begin{matrix}\left\{ \begin{matrix}{{H_{0}\text{:}\pi_{01}} = \pi_{11}} \\{{H_{1}\text{:}\pi_{01}} \neq \pi_{11}}\end{matrix} \right. & (162)\end{matrix}$

where π_(ij)=Pr(I_(d)=j|I_(d-1)=i). The null hypothesis states that theprobability of a breach on day d is the same regardless of whether abreach occurred on day d−1. The alternative is that a breach is less ormore likely to happen on day d if a breach occurred on day d−1. The teststatistic is the following likelihood ratio:

$\begin{matrix}{{LR_{ind}} = {{{- 2}\;\log\;\left( \frac{L\left( \Pi_{2} \right)}{L\left( \Pi_{1} \right)} \right)} = {{- 2}\;\log\;\left( \frac{\left( {1 - {\overset{\hat{}}{\pi}}_{2}} \right)^{n_{00} + n_{10}}{\overset{\hat{}}{\pi}}_{2}^{n_{01} + n_{11}}}{\left( {1 - {\overset{\hat{}}{\pi}}_{01}} \right)^{n_{00}}{{\overset{\hat{}}{\pi}}_{01}^{n_{01}}\left( {1 - {\overset{\hat{}}{\pi}}_{11}} \right)}^{n_{10}}{\overset{\hat{}}{\pi}}_{11}^{n_{11}}} \right)}}} & (163)\end{matrix}$

The term n_(ij) is designated as the number of observations with value ifollowed by j,

${{\hat{\pi}}_{01} = \frac{n_{01}}{n_{00} + n_{01}}},{{{and}\mspace{14mu}{\hat{\pi}}_{11}} = {\frac{n_{11}}{n_{10} + n_{11}}.}}$

Note that

${\overset{\hat{}}{\pi}}_{2} = \frac{n_{01} + n_{11}}{n_{00} + n_{01} + n_{10} + n_{11}}$

and that if the first observation is ignored, then

${\hat{\pi}}_{2} = {\hat{\pi} = \frac{n_{1}}{n_{0} + n_{1}}}$

as previously defined. The test statistic LR_(ind) is asymptoticallydistributed as the chi-square distribution χ² (1) with 1 degree offreedom as D→∞.

The p-value associated with the calculated test statistic LR_(ind)({circumflex over (π)}₂, {circumflex over (π)}₀₁, {circumflex over(σ)}₁₁) is computed as follows:

$\begin{matrix}{{p\_{value}}_{{LR}_{ind}} = {P{r\left( {{LR_{ind}} \geq {L{R_{ind}\left( {{\overset{\hat{}}{\pi}}_{2},{\overset{\hat{}}{\pi}}_{01},{\overset{\hat{}}{\pi}}_{11}} \right)}}} \right)}}} & (164)\end{matrix}$

The test can be performed at the 5% significance level and the followingpossible outcomes are considered:

Pass if the p-value computed using equation (164) is greater than orequal to 5%; and

Fail if the p-value is less than 5%.

The following R-A-G scale shown in Table 5 is preferred for reportingthe Christoffersen test results.

TABLE 5 R-A-G Threshold for Christoffersen Test R-A-G Threshold Gp-value ≥ 5% A 0.01% ≤ p-value < 5% R p-value < 0.01%

In the case of 2-day holding period, the Christoffersen test may be usedin combination with the non-overlapping aggregation method. The testprocedure is then applied separately to each of the h groups ofnon-overlapping time series, and the outcome is determined based on theaverage p-value taken across the groups.

There are cases when the test statistic in equation (163) may beundefined, most notably when n₁₁ is zero, i.e., when there are no twoconsecutive breaches. For 1-day holding period, a p-value of 1 may beformally reported and assigned Pass to the outcome of the test (or Greenon the R-A-G scale) in such cases. For 2-day holding period, a p-valueof 1 may be assigned to each of the groups for which the test statisticis undefined; however, the outcome may be determined based on thesmallest p-value across the groups in this case (rather than the averagep-value).

The Christoffersen test results for multiple portfolios can beaggregated using a method analogous to the overlapping/non-overlappingaggregation methods described above.

In general, the Christoffersen test tests the null hypothesis ofindependence against the first-order Markov property (rather than allforms of dependence in general). It is possible that the chi-squareapproximation may be inaccurate for small breach proportions(equivalently, high confidence levels) and only moderately large samplesizes. As a result, the actual significance level of the test may bemuch higher than the nominal specified significance level.

A purpose of the coverage ratio is to report the average IM coverage ofa particular portfolio, or group of portfolios, across time, regardlessof whether a breach has occurred or not. The coverage ratio may be usedto assess if IRM 1616 is acting in an excessively conservative manner.This test is distinct from passing the backtest, which could be achievedsimply by charging excessive margin.

The coverage ratio may be used by model calibration module 1628 in thecalibration process of IRM 1616. The coverage ratio may also be used,for example, for comparing the performance of IRM 1616 across differentportfolio groups, in different model configurations and/or to somebenchmarks.

The coverage ratio quantifies how much of a margin cushion that IRM 1616provides on average compared to realized VM. The coverage ratio (CR) iscalculated as:

$\begin{matrix}{{CR} = \left\{ \frac{\sum{IM}_{d}}{\sum{VM}_{d}} \right\}_{{VM_{d}} < 0}} & (165)\end{matrix}$

Here the summation runs over the backtest days for which the variationmargin is negative, i.e., VM_(d)<0.

A possible limitation might arise as a result of aggregating overmultiple backtest days. The coverage ratio may not provide informationabout how extreme the difference between IM_(d) and VM_(d) might be onany given day.

The breach fraction is a metric that may be used to calculate therealized coverage over the backtesting period for a particularportfolio.

The breach fraction for a portfolio is defined as follows:

$\begin{matrix}{{BF} = \frac{n_{1}}{D}} & (166)\end{matrix}$

The value of the breach fraction BF should be close to the coverageprobability p if the model coverage is correct.

Next, the breach severity metric is described. A purpose of the breachseverity metric is to capture and assess the average severity of theoccurred breaches of a single portfolio. In addition to the average, themaximum breach size of a single portfolio may also be provided.

Let {d₁, d₂, . . . , d_(u)} denote the set of dates where a breach eventoccurs and let BS_(d) be the breach size as defined in equation (152) onbacktest date d. The breach severity metric is defined as:

$\begin{matrix}{{BreachSeverity} = {\frac{1}{u}{\sum_{d \in {\{{d_{1},d_{2},\ldots\mspace{14mu},d_{u}}\}}}{BS_{d}}}}} & (167)\end{matrix}$

In addition to breach severity, the maximum breach size defined asmax(BS_(d))_(d∈{d) ₁ _(,d) ₂ _(, . . . ,d) _(u) _(}) may also beconsidered for further investigation. For the purpose of calculatingthese metrics, all days for which the Breach Size BS_(d) is undefineddue to the IM_(d) being equal to zero may be excluded.

In some examples, the Breach Severity values may be large due to the IMbeing small whereas the breaches may have low materiality (in terms ofthe number of portfolio ticks).

In an example portfolio evaluation matrix for testing IRM 1616, acrossall of the portfolios tested, IRM 1616 generates zero Basel TrafficLight “Red” instances, as well as an “Amber” count that is, depending onportfolio, either zero or low. The Real and Stylized Clearing Memberportfolios have zero instances of “Amber.” The observed number of Amberinstances in all cases is well below the number of Amber PortfoliosAllowed.

The conclusion of this test based on the metrics is that the coverageachieved by IRM 1616 is very likely to be 99% or greater across allportfolios and for the full time period backtested. It is noted thatsimilar results and similar conclusions have been determined for testingin the APC On configuration.

IRM 1616 passes the investigative Kupiec test, with zero instances of“Red” and, depending on the portfolio set, either zero or very fewinstances of “Amber.” The observed number of Amber instances for allportfolios is well below the number of Amber Portfolios Allowed. Theseresults are expected in view of the BTL results as shown above. Theconclusion is that with a high degree of likelihood, IRM 1616 is able toachieve 99% coverage or greater, based on the full backtest period.

A substantial portion of breach clusters tend to be event driven, wherea given macro event or meaningful announcement may cause IRM 1616 tobreach for more than one day in a row. Across Red and Amber portfolios,the majority of breach clusters are clusters of 2 sequential breachesonly, and the number of clusters does not exceed 3 in most cases.

From a model design perspective, there exists a tradeoff betweensufficient reactivity to avoid sequential breaches versus making IRM1616 overly reactive. IRM 1616 can be made more reactive, e.g. byreducing the Lambda parameter. However, increasing the reactivity may beundesirable from an overall margin stability or from ananti-procyclicality perspective.

Finally, to the extent significant macro events extend beyond a singlebusiness day, it can be expected that one or more Risk Factor orinstrument portfolios might breach for the duration in the same way thata single-day market shock creates a single day breach. This occurs inthe testing setup in a non-mitigated manner, that is, without theconsideration of intra-day IM calls.

In this example, sequential breach clustering remains a relativelyuncommon occurrence for IRM 1616: it occurs primarily for a narrow setof Risk Factor portfolios at identifiable macro-economic stress events.Overall Model performance is therefore deemed acceptable. Also, theresults suggest that IRM 1616 is generally able to “learn” from recentmarket events and is sufficiently reactive to prevent breach clusteringfor the significant majority of portfolios.

Procyclicality Testing Category

An initial margin model is desirably reactive to changes in marketconditions. However, it is undesirable for IRM 1616 to exhibit largestep changes in the IM, especially during periods of stress. Asspecified by applicable regulations, it is desirable for IRM 1616 tobehave in an anti-procyclical manner. To test the procyclicality, twocategories of metrics are used: primary and investigative metrics.

The primary metrics generally assess whether IRM 1616 is procyclicalbased on IM changes over a predetermined time period (in this example, a5-day horizon) during a pre-defined market period of stress. Morespecifically, the APC Expected Shortfall (APC ES) metric is evaluatedover the period B_(s) of the market stress defined as the 500-day periodfrom Sep. 3, 2007 to Aug. 10, 2009. This period is selected based on theaverage values of the EWMA volatility (generated with EWMA Lambda equalto 0.98) taken across all linear Risk Factors.

To compute the APC Expected Shortfall, the relative IM changes are firstcalculated as follows:

$\begin{matrix}{{\Delta\;{IM}_{{5D};d}} = {\frac{{IM}_{d}}{{IM}_{d - 5}} - 1}} & (168)\end{matrix}$

for d=6, 7, . . . , D. Then the APC Expected Shortfall metric iscalculated as

$\begin{matrix}{{ES_{APC}} = {E{S_{{5D};{95\%}}\left( {\Delta\;{IM}_{{5D};{6 \leq d \leq D}}} \right)}}} & (169)\end{matrix}$

where the ES_(5D; 95%)(⋅) is the Expected Shortfall operator at the95^(th) percentile level that computes the average of the largestrelative 5-day IM increases at the 95^(th) percentile level and higher.Note that the APC metric is concerned with relative IM increases, notrelative IM decreases.

To calculate the APC Expected Shortfall, all days are excluded for whichthe relative IM change ΔIM_(5D; d) is undefined due to the IM_(d-5)being equal to zero in rare cases.

Moreover, since the metric is affected by rolling to a new contract(causing the risk profile to change), the rolling dates are excludedfrom the metric computation. The exception is made for RI-CM Stylizedportfolios since excluding the rolling dates for these portfolios mayresult in only a small number of days left for the metric computation.

In this example, the calculation of the APC Expected Shortfall metric isbased on overlapping 5-day changes in the IM that exhibit serialdependency. Consequently, one large spike may affect 5 consecutivevalues of the APC Expected Shortfall. This may result in an increasederror of the Expected Shortfall operator. It is also noted that thereported changes may appear large as they are stated in percentageterms, but may be small in magnitude (i.e. have low materiality, in tickterms). For dynamic types of portfolios the IM can change due to changesin the portfolio risk profile and therefore the metric may not beapplied to such portfolios.

The investigative metrics, in general, complement the APC Expectedshortfall metric as there is not a single “industry-standard” method toquantify procyclicality. The metrics considered are the n-dayProcyclicality metric and Peak-to-Trough metric. These metrics arereferred to as BoE APC metrics.

For the n-day Procyclicality, let the n-day relative IM change bedenoted as:

$\begin{matrix}{{\Delta\;{IM}_{{nD};d}} = {\frac{{IM}_{d}}{{IM}_{d - n}} - 1}} & (170)\end{matrix}$

The n-day Procyclicality metric is then defined as the maximum of then-day relative IM changes over a given period of time:

$\begin{matrix}{{Procyclicality}^{(n)} = {\max\limits_{d}{\Delta\;{{IM}_{{nD};d}(}}}} & (171)\end{matrix}$

Similarly to the APC Expected Shortfall metric, the n-day Procyclicalitymetric is applied during period B_(S) of stress.

To calculate n-day Procyclicality, all days are excluded for which therelative IM change ΔIM_(nD; d) is undefined due to the IM_(d-n) beingequal to zero. The rolling dates of contracts are excluded from thecomputation, with an exception being made for RI-CM Stylized portfolios(by analogy with the APC Expected Shortfall computation).

The Peak-to-Trough metric is defined as the ratio of the maximum marginrequirement to the minimum margin requirement over a given period oftime:

$\begin{matrix}{{PeakToTrough} = \frac{\max\limits_{d}\left( {- {IM}_{d}} \right)}{\min\limits_{d}\left( {- {IM}_{d}} \right)}} & (172)\end{matrix}$

The Peak-to-Trough metric is applied during the period B_(S) of stress.To calculate this metric, all days are excluded for which IM_(d) isequal to zero (however, rolling dates are not excluded).

The n-day Procyclicality metric is obtained as the maximum of therelative IM changes and hence may be very sensitive to outliers. In someexamples, the Peak-to-Trough may be large even if the peak

$\max\limits_{d}\left( {- {IM}_{d}} \right)$

occurs prior to the trough

${\min\limits_{d}\left( {- {IM}_{d}} \right)};$

however, such cases do not present a concern since they correspond todecreases in margin requirements.

Regarding the portfolio evaluation matrix, note that the APC metrics maybe applicable to a subset of portfolios with static positions and slowlychanging risk profiles between expiry dates (relevant for RealInstruments). Assessing IRM 1616 for procyclicality includes carefulconsideration as it can be difficult to attribute margin changes to IRM1616 itself or to the risk profile changing due to the passage of time,or in the case of options when the underlying price diverges to, orconverges from, the strike level.

In this example of testing, in the group of RI-Stylized, the followingnon-option instrument portfolios are excluded from the APC assessment:portfolios with front contracts in STIR, Repo or SONIA/EONIA Futuressince such portfolios have rapidly changing risk profiles due tomonthly/quarterly cycle; and ERIS portfolios based on fixed expiriessince such portfolios have changing risk profiles when approachingexpiry.

In the groups of RI-Stylized and SI-Non-Linear Stylized, the optionportfolios included in the APC assessment are outright ATM optionportfolios where the underlying is not the front month expiry. ATMoutright option portfolios are included in the assessment since theoption portfolios were constructed based on relative strike, which couldcause risk profile jumps due to the reconstruction of the portfolio fortesting rather than IRM 1616 reacting to changes in volatilities (forexample). Option portfolios where the underlying is the front monthexpiry are excluded from the (example) assessment as the underlying hasa materially changing risk profile when approaching expiry.

RI-CM Real portfolios are excluded from the (example) APC assessment asthe portfolio risk profile is not held constant over time due topositions changing potentially on a daily basis. In the group of RI-CMStylized portfolios, all portfolios are included for the APC assessment.Since the APC metric results for these portfolios may be affected byrolling effects, such cases are carefully investigated and noted.

Overall, the testing results for the APC Expected Shortfall metric aredeemed acceptable for the Real Instrument portfolios, as well as for theRisk Factor and Synthetic Instrument portfolios. Thus, it may beconcluded that for all portfolios tested, IRM 1616 is able to generateInitial Margin during stress periods that is consistent with theexpectation of anti-procyclicality.

In addition to the procyclicality testing results based on thehistorical data provided above, the procyclicality results areillustrated for the two synthetic datasets. For the Benign syntheticdataset, the APC ES metric indicates zero instances of Red and very fewinstances of Amber. For the Regime Change synthetic dataset, there aresix Reds in total across the example 5,895 portfolios analyzed.

Sensitivity Testing Category

A purpose of sensitivity testing is to evaluate the change in modeloutputs due to a change in model inputs, parameter values, or marketconditions. Two types of sensitivity testing are described below:Rolling Backtest Analysis and Parameter Sensitivity Testing. The rollingbacktest analysis may be used to assess the performance of IRM 1616 overchanging market conditions, including periods of stress.

In this example, the test is conducted using a 500-day rolling lookbackwindow both in the APC On and APC Off configurations. The first lookbackwindow covers the period from Sep. 3, 2007 to Aug. 10, 2009. The windowis then shifted by one day with the last window covering the period fromMar. 10, 2016 to Feb. 14, 2018. A total of 2200 rolling windows areconsidered.

The backtesting allows assessment of the average performance of IRM 1616over a long backtesting period. The rolling backtest allows detectingwhether IRM 1616 under-performs in some time periods more than others.

The Basel Traffic Light Test is the Primary Metric that is applied tothe Rolling Backtest Analysis. The APC Expected Shortfall metric is alsoexamined. Each metric is evaluated for each of the lookback windows. Inthis example, the metrics are be applied to Real Instrument Stylizedportfolios.

In this example, the power of the statistical tests (such as the BTL)applied to each of the 500-day lookback windows is lower as compared tothe power of the tests applied in the full period backtesting due to asmaller sample size. This possible limitation may be mitigated by thefact that multiple (a total of 2200) lookback windows are considered.

The performance of IRM 1616 based on the BTL metric applied on a rolling500-day window is therefore deemed acceptable. A similar performance isalso determined when IRM 1616 is running in the APC on configuration.

For the parameter sensitivity testing analysis, the model inputparameters are modified by varying a given parameter value from itsbaseline level. A comparison is then made, holding all other parametersconstant at their Baseline values, of the backtesting output using thebacktesting metrics described below. Sensitivity testing is designed toassess the robustness of performance of IRM 1616 to changes in the inputparameters.

In this example, the parameter ranges are set to be relatively wide inorder to explore the stability of IRM 1616 for the different parametersettings. The explored parameter ranges can and do violate bothregulatory permitted parameter levels (e.g., holding period of one day,confidence level of less than 99%) as well as the parameterrecommendations and ranges (Lambda equal to one).

In a production environment, in some examples, IRM 1616 may beparameterized in a manner that meets regulatory requirements. However,from a research perspective it is important to understand the modelbehavior over a wide range of parameter values to understand modelstability to parameter selections in the neighborhood of (above andbelow) any values required by regulation.

The Basel Traffic Light Test, APC Expected Shortfall, and BreachFraction metrics may be applied to sensitivity testing. Note that all ofthese metrics may not be applied across all portfolios for sensitivitytesting.

In this example, the following Real Instrument portfolios are evaluatedas part of sensitivity testing:

-   -   Stylized Strategies: full period and rolling period testing is        conducted    -   Real CMs and Clients: full period is conducted for Basel Traffic        Light Test and Breach Fraction metric only.    -   Stylized CMs and Clients: full period for Basel Traffic Light        Test and Breach Fraction and stress period testing for APC        Expected Shortfall is conducted.

In this example, the model calibration process described above is notre-run when parameter changes are applied. Any dependencies betweenparameters that may impact the calibration process results are notconsidered.

Model sensitivity testing shows that IRM 1616 generally does not exhibita high degree of sensitivity to the set of parameter inputs that arebeing evaluated.

The EWMA Weight (Lambda) parameter leads to reduced IRM 1616 performancewhen the parameter is set outside the recommended bound to λ=1. Thenumber of both BTL Amber and Red instances increases with this setting.Similarly, when the Correlation Stress weight is reduced to zero, i.e.outside of the recommended range, the performance of IRM 1616 on the BTLmetric is somewhat deteriorated.

Incremental Testing Category

Incremental testing may help to evaluate the change in model outputs dueto the addition of model components. The progressive nature of thetesting allows for the isolation of potential model issues to aparticular component of IRM 1616. The testing starts with the most basicconfiguration and gradually adds additional model components. Comparisonof backtesting metrics across the configurations is performed. In thisexample, portfolios for the incremental testing include Real InstrumentStylized portfolios.

The following configurations show an example progression of adding modelcomponents:

-   -   Configuration A: The Volatility Forecast does not include the        Volatility Floor or Stress Volatility Component. APC treatment,        Correlation Stress Component, and Diversification Benefit        Component are not included in the IM calculation.    -   Configuration B (Configuration A+Stress Volatility        Component+APC): The Volatility Forecast accounts for the Stress        Volatility Component and APC Treatment.    -   Configuration C (Configuration B+Volatility Floor): The        Volatility Forecast accounts for the Volatility Floor.    -   Configuration D (Configuration C+CSC+Diversification Benefits):        The final IM calculation accounts for the Correlation Stress        Component and Diversification Benefit Component.

In this example, P&L rounding is not applied in the incremental testing,in order to better isolate the impact of other model components.Accordingly, the Risk Factor VM is used in the tests for consistency.This allows for the full period backtesting as there is no dependency onthe availability of the settlement VM for the instruments.

In this example, the Basel Traffic Light Test, Breach Fraction, and APCExpected Shortfall are applied to incremental testing.

The Lambda parameter is calibrated for the final configuration(Configuration D). In this example, the model calibration process is notre-run for each configuration.

Model Comparison with Historical Simulation Category

The Initial Margin for a VaR-based Historical Simulation model iscalculated for Real Instrument Stylized portfolios and compared to theIM of IRM 1616. Historical Simulation amounts to using unscaled returnsto generate the P&L distribution for the IM calculation. The length ofthe lookback window for Historical Simulation is set to W₀ days to matchthat of IRM 1616. In this example, time scaling is excluded from thistesting (e.g., 1-day MPOR is used) to allow for a more direct comparisonat a basic level between IRM 1616 and Historical Simulation.

In this example, the Basel Traffic Light Test, APC Expected Shortfall,and Breach Fraction metrics are used for the comparison.

Assumptions Backtesting Category

The impact of the chosen rounding treatment at the P&L level is comparedwith an alternative treatment which rounds the final IM rather than theP&L. The alternative rounding treatment is as follows. Consider aportfolio with N instruments, and let Q_(i), i=1, . . . , N denote thequantity of each instrument held, multiplied by the contract valuefactor, and let T_(i) denote the instrument minimum tick. Aftercalculating the intial margin, without P&L rounding, denoted byIM_(model), the following rounding treatment is applied:

$\begin{matrix}{{{IM}_{final} = {{- \left\lceil \frac{{IM}_{m{odel}}}{T} \right\rceil} \cdot \tau}},{\tau = {\max\limits_{{i = 1},\ldots\mspace{14mu},N}{{Q_{i}T_{i}}}}}} & (173)\end{matrix}$

where ┌⋅┐ denotes the ceiling operator that rounds the number up to thenearest integer. In particular, the method takes as its minimum tick thelargest one-tick movement of any of the individual positions held. It isdetermined that the two treatments produce similar results with respectto standard metrics such as BTL and coverage ratio, based on the periodfrom Sep. 3, 2007 to Feb. 14, 2018.

The chosen FX conversion method is compared with an alternative FXconversion method in which the spot FX rate (i.e., the FX Rate on theValue Date) is used. No scaling of the FX Risk Factors takes place inthis case. Overall it is determined that the two FX conversion methodsgenerate close results. Specifically, across all 21 multi-currencyportfolios in the set of RI-Stylized portfolios, the 99^(th) percentileof absolute IM discrepancy |IM₁−IM₂| is below 1 portfolio tick (definedas the largest instrument tick across all portfolio instruments) in theAPC Off, 2-day holding period configuration (based on the backtestperiod from Sep. 3, 2007 to Feb. 14, 2018).

The initial margin of option portfolios is examined for which realizedlosses cannot exceed the premium (in magnitude). Specifically, it isinvestigated whether the magnitude of the initial margin for suchportfolios exceeds the magnitude of the premium. It is expected that theexceedances may be present due to such features of IRM 1616 as theCorrelation Stress Component and P&L Rounding. However, such exceedancesmay be acceptable when they are no larger than 1-2 relevant tick sizes.

Backtesting of option portfolios is performed in the set of RealInstrument Stylized portfolios over the period from Sep. 3, 2007 to Feb.14, 2018. Those portfolios are examined for which realized losses do notexceed the premium and the premium is negative (i.e., there is a cost toconstruct the portfolio). In this example, the portfolios includeoutright options and option strategies, in particular box spread,conversion, straddle, strangle, call-put spread, butterfly, calendarspread and calendar straddle strategies.

The chosen time scaling method in which the holding period multiplier isequal to the square root of the holding period is compared with the timescaling method in which the holding period multiplier is equal to themaximum of the AR(1) time scaling factor and the square root of theholding period. In this example, backtesting of the RI-Stylizedportfolios is performed in the APC Off, 2-day holding periodconfiguration over the period from Sep. 3, 2007 to Feb. 14, 2018.Overall it is determined that the two methods generate similar results.Specifically, in cases where the median relative IM difference (takenacross all backtest dates for a given portfolio) exceeds 3%, the(absolute) IM difference does not exceed 2 portfolio ticks.

Other approaches to time-scaling may include those based on usingnon-overlapping or overlapping h-day returns. (In these casestime-scaling is implicit and can be referred to as time aggregation.)However, such approaches may have limitations of their own. For example,the use of non-overlapping returns may result in serious reduction inthe sample size (from W to └W/h┘). Using overlapping returns maygenerate profound serial dependency in the data even if it wasoriginally independent. Accordingly, the adopted time scaling approachmay be preferred over such time-aggregation methods.

Example LRC Model

Next, an example configuration and testing of LRC model 1618 isdescribed. The testing of LRC model 1618 may be performed, in someexamples, by model testing module 1630. In general, LRC model 1618include a number of parameters that may be configured for LRC model 1618to operate according to predetermined performance characteristics. Insome examples, LRC model 1618 may be configured by selectingRepresentative Instruments, Liquidity Buckets, CC and BAC groups. Insome examples, parameters of LRC model 1618 may be periodically examinedand/or adjusted.

The use of the Concentration Charge/Bid-Ask Charge groups in LRC model1618 allows assessment of the liquidity risk more appropriately amonginstruments with common attributes, whereby market liquidity is expectedto be different between different groups. Where the exposure forRepresentative Instruments is offsetting in the same group, this mayprevent Clearing Members from having to post margins in excess of theeconomically relevant amount.

In the Baseline Model configuration, instruments that share the ProductCategory and Currency of the underlying instrument are assigned to thesame group.

Liquidity Buckets collect instruments from one or more Expiry Bucketsthat can be represented by a single instrument, i.e. RepresentativeInstrument. This allows the replacement of instrument positions in theLiquidity Bucket with a single position of the RepresentativeInstrument. The Liquidity Buckets selection may take into considerationhow products are traded, e.g., STIR packs (0-1Y, 1-2Y, etc.), and acorrelation assessment of returns between each pair of instrumentswithin a Liquidity Bucket. Instruments belonging to the same LiquidityBucket are expected to exhibit high levels of correlation in thereturns.

The Representative Instrument for each Liquidity Bucket may be selectedbased on multiple considerations, including trading activity andstability of the risk profile. Specifically, it is desirable for aRepresentative Instrument to be actively traded (relative to otherinstruments in the Liquidity Bucket); at the same time, the risk profileof the Representative Instrument should be stable. In view of thestability regulations, in this example, a linear instrument is selectedas a Representative Instrument. To assess what instruments are activelytraded, the average daily volume (ADV) is generally used, computed asfor a given instrument based on all available historical data.

For STIR futures, a contract is generally selected with the greatest ADVas the Representative Instrument across the quarterly contracts in aLiquidity Bucket. Typically, for the 2Y+Liquidity Buckets, a trend ofdecreasing daily volume is observed, which means that theshortest-time-to-expiry quarterly contract per bucket is selected as RI.For the 1Y Liquidity Bucket, the second-quarter contract is selected asthe RI, which generally has a higher ADV than the first-quartercontract, although trading activity dominance does fluctuate across allfour quarterly contracts in-between rolling cycles.

For Repos, in this example, the first-month contracts (1M) are notselected as the Representative Instrument because these contracts areessentially averaging products and may not be a good representative ofall expiries. Among the remaining expiries, the contract with thehighest ADV is selected, e.g., the second-month contract (2M).

For GBP Bond futures, the first-quarter contract is selected, in thisexample, as the Representative Instrument for each Liquidity Bucketbased on the ADV considerations. For EUR Bond futures, due to low ADVvalues that have been observed, the first-quarter contract is selectedas the Representative Instrument in line with our selection for GBP Bondfutures.

For EUR Swapnotes, the first-quarter contract is selected as theRepresentative Instrument for each Liquidity Bucket because thesecontracts typically have the highest ADV (relative to the instruments inthe Liquidity Bucket) and do not typically pose concern for stability ofthe risk profiles (e.g., in contrast with the first quarter STIRcontracts). For GBP and USD Swapnotes, due to low ADV values that havebeen observed, the first-quarter contract is selected as theRepresentative Instrument in line with our selection for EUR Swapnotes.

For ONIA, the second-month contract is selected, in this example, as theRepresentative Instrument because the first-month contract may have anunstable risk profile. For ERIS, the first-quarter on-the-run contractis selected for each of the 2, 5, 10 and 30 year tenors for GBP and EURas the Representative Instruments. The selection, in this example, isbased on the relative trading activity of the front-quarter contracts ascompared to other expiries. It is understood that RepresentativeInstrument and Liquidity Bucket selections may be subject to periodicreviews.

The Minimum Delta Δ_(% min) acts as a floor on the magnitude of thetheoretical delta of an option. The intent of this floor is to avoidmissing zero delta or near-zero delta options that nonetheless may stillpose a risk to the CCP. Imposing a floor on option deltas may ensurethat deep out-of-the-money options that may appear near-riskless to theVaR method for portfolio representation are captured by the calculationin the Delta method. On the Bid-Ask Charge side, the Minimum Delta helpsto avoid instances where certain option portfolios may produce a zerocharge under either method, whilst in a close-out process they may stillattract a cost due to a lack of liquidity in deep out-of-the-moneyoptions.

To illustrate, consider a simple portfolio of one deep out-of-the-moneyoption that maps to a Representative Instrument, with Δ≈0, whose P&Lvector entries are also near zero. Consequently, BAC^((V))≈0 andBAC^((D))≈Δ_(% min)·BAC_(r).

The Minimum Delta value is selected based on the followingconsiderations. The VaR method for portfolio representation may not beable to capture options with the probability of expiring in-the-moneybelow 1% (since the confidence level in the VaR method is set to 99%).The value of Minimum Delta Δ_(% min) may be selected so as to ensureprotection for deep out-of-the-money options. Specifically, in thenormal model for option pricing, the magnitude of option delta is equalto the probability of the option to expire in-the-money; in thelognormal model, this is approximately true assuming the time-to-expiryis relatively short.

The Concentration Thresholds (CTs), set separately per Liquidity Bucketfor each Representative Instrument, may determine the capacity or volumethat may be traded in one day with no impact on market pricing. Forcalibration of CT, volume data and open interest data are used. Thecalibration process for the Concentration Thresholds may depend on anumber of parameters. The calibration methodology of the CTs and thevalue selection process for the parameters involved in the calibrationare discussed below.

Example Calibration of LRC Model

In the example below, the CT is calculated by reference to a ShortLookback Window of the most recent W_(S) business days and to a LongLookback Window of the most recent W_(L) business days. The CT for eachRepresentative Instrument is calibrated using the following hierarchicalapproach:

-   -   1. For each day in the Lookback Window, the maximum volume        V_(max) across the futures instruments within the Liquidity        Bucket is extracted. Similarly, the sum V_(sum) of the volumes        across the futures instruments within the same Liquidity Bucket        is extracted.    -   2. Compute V=max(w_(sum) ^(v)·V_(sum), V_(max)) for each day in        Long Lookback Window where w_(sum) ^(v) is a configurable        parameter set in a range between 0 and 100%.    -   3. Compute {circumflex over (V)}_(S) and {circumflex over        (V)}_(L) as the averages of V over the Short and Long Lookback        Windows, respectively, based on the values obtained in the        previous step. Days with zero volume are included in the        averaging process, except for exchange holidays.    -   4. Compute the average volume AV=max ({circumflex over (V)}_(S),        {circumflex over (V)}_(L)).    -   5. Repeat steps 1 through 4 for the open interest data instead        of volume. In step 2 use the configurable parameter w_(sum)        ^(OI) (in place of w_(sum) ^(v)). The execution of these steps        results into the average open interest AOI.    -   6. Compute the Concentration Threshold as

$\begin{matrix}{{CT} = {\max\left( {{\beta_{AV} \cdot {AV}},{\beta_{AOI} \cdot {AOI}},{floor}_{CT}} \right)}} & (174)\end{matrix}$

-   -    The CT is floored to avoid pushing the Concentration Charge to        arbitrarily large values. This amount represents a small        position and is deemed not concentrated for all products in        scope.

In this example, volume and open interest of option instruments are notconsidered in the Baseline configuration. However, in some examples,they may also be taken into account in steps 1 and 4 above.

Adjustments may be applied to CT in instances where model output isdeemed to be unrepresentative of current liquidation risk faced by theCCP. Such adjustments are expected to occur on an exception basis ratherthan as a rule to be frequently applied. If LRC model 1618 isconsistently overruled, this may serve as a guidepost to review theproper functioning and design of the model.

Parameter w_(sum) ^(V) measures what proportion of the trades isdirectional versus trading in spreads/strategies. The parameter was setto 45%, in this example, based on the empirical analysis of exchangedata. The interpretation and value selection for parameter w_(sum) ^(OI)is analogous to that of w_(sum) ^(V).

The value of parameter floor_(CT) is set, in this example, to 100 lots.Generally this is considered to be an amount that can be liquidated,without any market impact, in one day.

The Short Lookback Window Length is set, in this example, to 60 businessdays, which approximately corresponds to a calendar period of 3 months.The averaging over the Short Lookback Window is performed to be able tocapture more recent changes in liquidity, such as the increase ofliquidity at the end of 2015. The Long Lookback Window Length is set, inthis example, to 250 business days, which approximately corresponds to acalendar period of 1 year. The averaging over the Long Lookback Windowis performed to ensure the stability of the estimated volume from oneday to the next.

It should be noted that the CT calibration with its parameter choices,in some examples, may be determined under a conservative assumption ofconsidering ETD volume and OI only, thus discounting entirely OTC volumeand OI.

Example Bid-Ask Parameters of LRC Model

Due to the nature of bid and ask prices, which may lead to an executedtrade or just an open order to trade, the determination of the Bid-Askspread parameters (BA_(r)) requires a careful consideration. The Bid-Askspread parameterization process follows a waterfall structure thatstarts with observed Bid-Ask spread data reported by one or moreelectronic exchange or one or more third party systems followed byfeedback from one or more systems associated with risk and/orproduct/market experts. In some examples, the available data may bereviewed and used to determine the suitability of the Bid-Ask chargesfor the purposes of LRC model 1618, and may provide for any adjustmentswhere required.

Bid-ask parameters' calibration may use available data that is sourcedas discussed above. In this example, the Bid-Ask spread estimate foreach Representative Instrument is calculated as follows:

-   -   1. Extract the most recent quarter of data that follows the        requirements of MiFID II Best Execution Report.    -   2. For each day, group instruments into their Liquidity Buckets        and calculate the minimum, average and maximum of each of the        following two fields in the report:        -   i. AVG_EFFECTIVE_SPREAD        -   ii. AVG_SPREAD_AT_BEST_BID_AND_OFFER    -   3. Calculate the (equally weighted) average of each of the        estimates in Step 2 over the available days.

Here:

-   -   AVG_EFFECTIVE_SPREAD is the one-day average of an instrument's        bid-ask spreads where each spread is the difference between best        bid/offer before a trade is executed (excluding the order that        caused the trade). The spread is calculated on every executed        only trade and only if the both best bid/offer existed before        the trade.    -   AVG_SPREAD_AT_BEST_BID_AND_OFFER is the one-day average of an        instrument's bid-ask spreads, where each spread is calculated on        every new order entry (when the new order is already in the        order book) if both best offer and best bid exist. The orders        that cause an executed trade are recorded as zero spreads and        used in the averaging.

In this example, the AVG_EFFECTIVE_SPREAD field is used as the defaultfor the Bid-Ask spread parameter setting, as this incorporates the bestbid and best offer in the market prior to executed trades. Where theAVG_EFFECTIVE_SPREAD field returns null (no trades), theAVG_SPREAD_AT_BEST_BID_AND_OFFER filed is used as a fallback, ifavailable.

Relevant proxy of parameters may be used in the following cases:

-   -   a) For products that are deemed to be liquid in the interest        rate market, e.g. EuroDollar futures, but for which there is no        reported trading activity, the Bid-Ask spread parameters can be        set based on third-party sources or set to be 1 minimum price        increment of the instrument. The parameters will be reviewed        once the market becomes actively traded.    -   b) For instruments with little or no reported trading activity,        the Bid-Ask spread parameters can be proxied using the nearest        tenor or look alike products. The parameters will be reviewed        once the market becomes actively traded.        The final parameters set by the Clearing Risk Department, taking        into consideration the estimates detailed above, may be reviewed        further, considering the instruments' trading behavior, for        example, during expiries when trading activity rolls to new        expiries.

Model Testing Module with respect to the LRC Model

The Liquidity Risk Charges for different portfolios over a testingperiod are examined for various calibration setups. A number ofempirical tests are performed to assess the performance of the LRCmodel.

The testing plan may be separated into the following main categories:

-   -   1. Stability Testing: to assess the stability of CC and BAC by        considering them together with IRM 1616.    -   2. Sensitivity Testing: to assess the change in CC and BAC        values due to a change in a parameter or configuration of LRC        model 1618.        Additionally, a liquidation period analysis is performed, as        discussed below.

Each test period provides a statistically meaningful sample and coversat least one financial market stress event. Notably, each test periodincludes stress observed following the United Kingdom's referendumresult, in favour of leaving the European Union, on 24 Jun. 2016.Additionally, in December of 2015, the ECB cut the deposit rate by 10bps and expanded the QE program, a decision the market did not expect.The testing period is constrained by the available data for actualClearing Member portfolios and the raw data required for modelparameterization, e.g. volume.

In each of the tests involving the CC, either a changing or staticConcentration Threshold is used. The changing CT is calibrated asdiscussed above. In the case of the static CT, a single value of theConcentration Threshold CT_(r) for each Representative Instrument isused throughout the testing period. This value is calibrated based onthe 1-year period up to 22 Jan. 2018 (which is the last CT calibrationdate prior to the test period end date of 14 Feb. 2018).

A static value of the Bid-Ask spread BA_(r) for each RepresentativeInstrument is used throughout the testing period in all tests. It iscalibrated based on the MiFID II Best Execution Report published in thesecond quarter of 2018 and covering the first quarter of 2018.

In this example, the following portfolio types are considered: real CM,stylized CM and synthetic stylized. For the real CM type, positions asobserved historically in the accounts of Clearing Members. For thestylized CM type, theoretical instrument portfolios are designed basedon the concentrated positions of CMs. For the synthetic stylized type,stylized theoretical portfolios with varying levels of concentration aredesigned, in particular, to test the sensitivity of CC with respect toposition size.

Stability testing of CC and BAC is performed. The stability of LRC model1618 is a key factor in assessing its performance. Since LRC model 1618may be an add-on charge for IRM 1616, the stability of the LRCcomponents may be performed by considering the components together withthe IM determined by IRM 1616. Intuitively, if an account's risk profileremains unchanged from one day to the next, the sum of the LRC outputand the IM (of IRM 1616) should also be stable or more importantly notprocyclical.

The stability of the output of LRC model 1618 may be assessed, in anexample, using the 5-day APC Expected Shortfall (APC ES) at the 95^(th)percentile level in line with the procyclicality assessment.Specifically, to assess the stability of a charge C_(d) on day d (C_(d)can be taken to be, for example, the sum of CC and the initial margin(from IRM 1616), the APC ES is computed as follows.

First the relative 5-day changes are calculated:

$\begin{matrix}{{\Delta C_{{5D};d}} = {\frac{C_{d}}{C_{d - 5}} - 1}} & (175)\end{matrix}$

for d=6, 7, . . . , D where D is the total number of days for which thecharge is computed. Then the 5-day APC Expected Shortfall metric iscalculated as

$\begin{matrix}{{ES_{APC}} = {E{S_{{5D};{95\%}}\left( C_{{5D};{6 \leq d \leq D}} \right)}}} & (176)\end{matrix}$

where the ES_(5D; 95%) is the Expected Shortfall operator at the 95^(th)percentile level that computes the average of the largest relative 5-dayincreases at the 95^(th) percentile level and higher. Note that the APCES metric is concerned with relative increases, not relative decreases.

To calculate the APC ES, all days are excluded for which the relativechange is undefined due to the C_(d-5) being equal to zero (in very rarecases). Moreover, since the APC ES metric is affected by rolling to anew contract in the construction of historical stylized portfolios(causing the risk profile to change), the rolling dates are excludedfrom the metric computation.

One of three possible outcomes on the R-A-G scale is assigned as shownin Table 6 below:

TABLE 6 The outcomes for the APC ES Metric on the R-A-G scale R-A-G APCES Outcome ES_(APC) Thresholds G Low Procyclicality  <50% A MediumProcyclicality >=50% and <=100% R High Procyclicality >100%

Note that the stability assessment may be performed only for a subset ofportfolios with static positions and slowly changing risk profilesbetween expiry dates. Assessing LRC model 1618 for stability may includecareful consideration as it can be difficult to attribute changes to LRCmodel 1616 itself or to the risk profile changing due to the passage oftime, or in the case of options when the underlying price diverges from,or converges to, the strike level.

In this example, for the stability testing, two portfolio sets are used:Stylized CM and Synthetic Stylized portfolio sets. In the SyntheticStylized set, a subset of 72 portfolios are considered that conform tothe desired outcomes (discussed above).

The stability of the Concentration Charge and Bid-Ask Charge may beassessed, in an example, by applying the 5-day APC ES metric to thefollowing four (4) charges: the initial margin (IM) determined by IRM1616); IRM 1616IM+CC; IRM 1616IM+BAC; and IRM 1616IM+CC+BAC.

Next, sensitivity testing is discussed. A purpose of sensitivity testingis to evaluate the change in model outputs due to a change in parametervalues or configuration settings. For CC, sensitivity is tested withrespect to Concentration Thresholds, to CCG selection, to a differentLiquidity Bucket selection, to the size of portfolio positions, and tothree model parameters. For BAC, sensitivity is tested with respect toBACG selections, to a different Liquidity Bucket selection, and to theMinimum Delta parameter.

In this example, sensitivity of CC and BAC to grouping configurations istested using the set of 17 Stylized CM portfolios. Both for CCG andBACG, two grouping configurations are considered, in addition to theBaseline configuration. From most to least granular the configurationsare:

-   -   Grouping configuration 1 (GC1): All instruments with the same        currency are grouped together. This produces 4 different groups        (CHF, EUR, GBP, and USD).    -   Grouping configuration 2 (GC2): All instruments are assigned to        the same group.

This results into a single group (Rates).

The expectation is that the more granular the grouping is, the lessportfolio benefit will be provided between positions when calculatingthe charges. Accordingly a more granular grouping would generally resultin a higher CC and BAC.

The sensitivity of CC as well as BAC are tested with respect to theLiquidity Bucket selection using the set of 261 Synthetic Stylizedportfolios. In this example, two configurations are considered: theBaseline configuration as discussed above, and an alternativeconfiguration.

Next, the sensitivity of CC is tested with respect to the change in thesize of portfolio positions using the example set of 261 StylizedSynthetic portfolios. In this example, the portfolio set includesoutrights as well as spreads/butterflies. Each portfolio falls into oneof three categories: mildly concentrated, highly concentrated, andextremely concentrated. The size of each position in a mildly, highlyand extremely concentrated portfolio is selected so that thecorresponding Liquidation Period is 0.5 days, 2 days and 6 days,respectively. The size of all positions does not change over time,except for a subset of delta-hedged portfolios.

It is expected that the CC generally increases as the portfolio becomesmore concentrated, given the portfolio is structurally the same and onlythe size of the position is changing.

Next, the sensitivity of CC is tested with respect to the ConcentrationThresholds using the example set of 248 Real CM accounts. Specificallyin addition to the Baseline configuration, two example alternativeconfigurations are considered in which the Concentration Threshold foreach Representative Instrument is obtained by multiplying the Baselinevalue of CT by a factor of 0.5 and a factor of 2, respectively.

As expected, the CC generally increases as the CT decreases. Moreover,the CC increases non-linearly, i.e., at a higher rate for a smaller CT.

Next, the sensitivity of CC and BAC are tested with respect to thefollowing parameters: w_(sum) ^(V) and w_(sum) ^(OI), β_(AOI), andMinimum Delta. In this example, the set of Synthetic Stylized portfoliosis used.

Next, a sensitivity of CC to β_(AOI) is discussed. As expected,decreasing (respectively, increasing) the value of β_(AOI) generallyleads to a greater (respectively, smaller) CC.

Next, the sensitivity of CC to w_(sum) ^(V)/w_(sum) ^(OI) values istested using the Synthetic Stylized portfolio set. As expected,decreasing (respectively, increasing) the value of w_(sum) ^(V)/w_(sum)^(OI) generally leads to a greater (respectively, smaller) CC.

Next, the sensitivity of CC and BAC to the Minimum Delta values istested using 192 option portfolios of the Synthetic Stylized set. Asexpected, increasing (respectively, decreasing) the value of MinimumDelta generally leads to a greater (respectively, smaller) CC. For theBAC, it is observed that it generally exhibits little sensitivity to theconsidered Minimum Delta values. Specifically, in this example, theaverage ratio of IM+BAC in each of the two alternative configurations tothe IM+BAC in the Baseline configuration remains within the intervalfrom 99.8% to 100.2% on all dates in the test period.

Next, a liquidation period analysis is discussed. In this example, thelength of each liquidation period LP_(r) is analyzed using the set ofReal CM portfolios over the testing window from 26 Jan. 2015 to 14 Feb.2018. In the analysis, the average for each LP_(r) is computed acrossall portfolios and all dates considered.

Computer Architecture

Systems and methods of the present disclosure may include and/or may beimplemented by one or more specialized computers or other suitablecomponents including specialized hardware and/or software components.For purposes of this disclosure, a specialized computer may be aprogrammable machine capable of performing arithmetic and/or logicaloperations and specially programmed to perform the functions describedherein. In some embodiments, computers may comprise processors,memories, data storage devices, and/or other commonly known or novelcomponents. These components may be connected physically or throughnetwork or wireless links. Computers may also comprise software whichmay direct the operations of the aforementioned components. Computersmay be referred to with terms that are commonly used by those ofordinary skill in the relevant arts, such as servers, personal computers(PCs), mobile devices, and other terms. It will be understood by thoseof ordinary skill that those terms used herein are interchangeable, andany special purpose computer specifically configured to perform thedescribed functions may be used.

Computers may be linked to one another via one or more networks. Anetwork may be any plurality of completely or partially interconnectedcomputers wherein some or all of the computers are able to communicatewith one another. It will be understood by those of ordinary skill thatconnections between computers may be wired in some cases (e.g., viawired TCP connection or other wired connection) and/or may be wireless(e.g., via a WiFi network connection). Any connection through which atleast two computers may exchange data can be the basis of a network.Furthermore, separate networks may be able to be interconnected suchthat one or more computers within one network may communicate with oneor more computers in another network. In such a case, the plurality ofseparate networks may optionally be considered to be a single network.

The term “computer” shall refer to any electronic device or devices,including those having capabilities to be utilized in connection withone or more components of risk engine architecture 100 (FIG. 1) and riskmanagement system 1600 (FIG. 16) (including components 1610-1630 of riskengine 1602, data source(s) 1604, data recipient(s) 1606) or an externalsystem, such as any device capable of receiving, transmitting,processing and/or using data and information. The computer may comprisea server, a processor, a microprocessor, a personal computer, such as alaptop, palm PC, desktop or workstation, a network server, a mainframe,an electronic wired or wireless device, such as for example, a cellulartelephone, a personal digital assistant, a smartphone, or any othercomputing and/or communication device.

The term “network” shall refer to any type of network or networks,including those capable of being utilized in connection witharchitecture 100 and system 1600 described herein, such as, for example,any public and/or private networks, including, for instance, theInternet, an intranet, or an extranet, any wired or wireless networks orcombinations thereof.

The term “computer-readable storage medium” should be taken to include asingle medium or multiple media that store one or more sets ofinstructions. The term “computer-readable storage medium” shall also betaken to include any medium that is capable of storing or encoding a setof instructions for execution by the machine and that causes the machineto perform any one or more of the methodologies of the presentdisclosure.

FIG. 28 illustrates a functional block diagram of a machine in theexample form of computer system 2800 within which a set of instructionsfor causing the machine to perform any one or more of the methodologies,processes or functions discussed herein may be executed. In someexamples, the machine may be connected (e.g., networked) to othermachines as described above. The machine may operate in the capacity ofa server or a client machine in a client-server network environment, oras a peer machine in a peer-to-peer (or distributed) networkenvironment. The machine may be any special-purpose machine capable ofexecuting a set of instructions (sequential or otherwise) that specifyactions to be taken by that machine for performing the functionsdescribe herein. Further, while only a single machine is illustrated,the term “machine” shall also be taken to include any collection ofmachines that individually or jointly execute a set (or multiple sets)of instructions to perform any one or more of the methodologiesdiscussed herein. In some examples, one or more components of riskengine architecture 100 or risk engine 1600 (e.g., components 1610-1630)may be implemented by the example machine shown in FIG. 28 (or acombination of two or more of such machines).

Example computer system 2800 may include processing device 2802, memory2806, data storage device 2810 and communication interface 2812, whichmay communicate with each other via data and control bus 2818. In someexamples, computer system 2800 may also include display device 2814and/or user interface 2816.

Processing device 2802 may include, without being limited to, amicroprocessor, a central processing unit, an application specificintegrated circuit (ASIC), a field programmable gate array (FPGA), adigital signal processor (DSP) and/or a network processor. Processingdevice 2802 may be configured to execute processing logic 2804 forperforming the operations described herein. In general, processingdevice 2802 may include any suitable special-purpose processing devicespecially programmed with processing logic 2804 to perform theoperations described herein.

Memory 2806 may include, for example, without being limited to, at leastone of a read-only memory (ROM), a random access memory (RAM), a flashmemory, a dynamic RAM (DRAM) and a static RAM (SRAM), storingcomputer-readable instructions 2808 executable by processing device2802. In general, memory 2806 may include any suitable non-transitorycomputer readable storage medium storing computer-readable instructions2808 executable by processing device 2802 for performing the operationsdescribed herein. For example, computer-readable instructions 2808 mayinclude operations performed by components 1610-1630 of risk engine1602), including operations shown in FIGS. 19-25 and 27). Although onememory device 2806 is illustrated in FIG. 28, in some examples, computersystem 2800 may include two or more memory devices (e.g., dynamic memoryand static memory).

Computer system 2800 may include communication interface device 2812,for direct communication with other computers (including wired and/orwireless communication) and/or for communication with a network. In someexamples, computer system 2800 may include display device 2814 (e.g., aliquid crystal display (LCD), a touch sensitive display, etc.). In someexamples, computer system 2800 may include user interface 2816 (e.g., analphanumeric input device, a cursor control device, etc.).

In some examples, computer system 2800 may include data storage device2810 storing instructions (e.g., software) for performing any one ormore of the functions described herein. Data storage device 2810 mayinclude any suitable non-transitory computer-readable storage medium,including, without being limited to, solid-state memories, optical mediaand magnetic media.

All exemplary embodiments described or depicted herein are providedmerely for the purpose of explanation and are in no way to be construedas limiting. Moreover, the words used herein are words of descriptionand illustration, rather than words of limitation. Further, althoughreference to particular means, materials, and embodiments are shown,there is no limitation to the particulars disclosed herein.

1. A system for efficiently modeling datasets, the system comprising:one or more processors executing machine-readable instructions stored ina non-transitory storage medium, thereby causing the system to: receiverisk factor data and additional data associated with one or morefinancial portfolios; generate a buffered margin by applying acorrelations stress component to an initial margin for the one or morefinancial portfolios to account for sudden increases or decreases in therisk factor data; determine a portfolio level liquidity risk for the oneor more financial portfolios based on the additional data and thebuffered margin; execute one or more assessment processes on theportfolio level liquidity risk to account for price movements; andexecute one or more assessment processes on the portfolio levelliquidity risk to account for market volatility.
 2. The system of claim1, further comprising machine-readable instructions that, when executedby the one or more processors, further cause the system to: execute afiltered historical simulation process comprising applying a scalingfactor to historical pricing data for the risk factor data to resemblecurrent market volatility.
 3. The system of claim 2, further comprisingmachine-readable instructions that, when executed by the one or moreprocessors, further cause the system to: generate portfolio profit andloss values for the one or more financial portfolios based on results ofthe risk factor simulation process, wherein the portfolio profit andloss values are used to determine the initial margin.
 4. The system ofclaim 2, further comprising machine-readable instructions that, whenexecuted by the one or more processors, further cause the system to:retrieve the historical pricing data for the risk factor data; determinestatistical properties of the historical pricing data; and performde-volatilization and re-volatilization of the historical pricing datato adjust the historical pricing data for the current market volatility.5. The system of claim 2, further comprising machine-readableinstructions that, when executed by the one or more processors, furthercause the system to: execute a volatility forecast comprising avolatility floor configured to adapt to current market environmentconditions.
 6. The system of claim 5, wherein the volatility forecastcomprises a stress volatility component associated with market stressperiods.
 7. The system of claim 5, wherein the volatility forecastincludes an anti-pro-cyclicality component (APC) configured to mitigatepro-cyclicality risk.
 8. The system of claim 3, wherein the systemgenerates the portfolio profit and loss values by executingmachine-readable instructions that cause the system to: generate one ormore risk factor scenarios based on the results of the risk factorsimulation process; generate one or more instrument pricing scenariosbased on the one or more risk factor scenarios; generate one or moreprofit and loss scenarios at an instrument level, based on the one ormore instrument pricing scenarios; and aggregate the one or more profitand loss scenarios at the instrument level to form one or more profitand loss scenarios at a portfolio level.
 9. The system of claim 1,further comprising machine-readable instructions that, when executed bythe one or more processors, further cause the system to: apply aportfolio diversification benefit to the initial margin, the portfoliodiversification benefit comprising a predetermined benefit limit. 10.The system of claim 1, further comprising machine-readable instructionsthat, when executed by the one or more processors, further cause thesystem to: determine a concentration charge and a bid-ask charge basedon one or more equivalent portfolio representations of the one or morefinancial portfolios; and determine the portfolio level liquidity riskbased on the combination of the concentration charge and the bid-askcharge.
 11. The system of claim 10, wherein the one or more equivalentportfolio representations comprise a first representation based on adelta technique and a second representation based on a value-at-risk(VaR) technique.
 12. The system of claim 1, further comprisingmachine-readable instructions that, when executed by the one or moreprocessors, further cause the system to: generate one or more syntheticdatasets configured to model at least one of a benign condition and aregime change condition.
 13. The system of claim 1, further comprising amargin model is defined by the machine-readable instructions andexecuted by the one or more processors, said margin model configured togenerate the buffered margin is generated.
 14. The system of claim 1,further comprising a liquidity risk charge (LRC) model defined by themachine-readable instructions and executed by the one or moreprocessors, said LRC model configured to determine the portfolio levelliquidity risk and the execute the at least one assessment process isperformed.
 15. The system of claim 14, further comprisingmachine-readable instructions that, when executed by the one or moreprocessors, further cause the system to: test one or more of the marginmodel and the LRC model according to one or more testing categories. 16.The system of claim 15, wherein the one or more testing categoriescomprise one or more of fundamental characteristics, backtesting,pro-cyclicality, sensitivity, incremental addition of one or more modelcomponents, model comparison with historical simulation, and assumptionbacktesting.
 17. The system of claim 1, wherein the one or morefinancial portfolios comprise one or more financial products and one ormore currencies.
 18. The system of claim 17, further comprisingmachine-readable instructions that, when executed by the one or moreprocessors, further cause the system to: apply a currency allocation tothe initial margin across the one or more currencies.
 19. The system ofclaim 17, wherein the one or more financial products comprise one ormore of a non-linear financial product and a linear financial product.20. The system of claim 19, further comprising machine-readableinstructions that, when executed by the one or more processors, furthercause the system to: empirically model the non-linear financial productand the linear financial product by a same empirical modeling process.